Optimized stochastic resonance signal detection method

ABSTRACT

Apparatus and method for detecting micro-calcifications in mammograms using novel algorithms and stochastic resonance noise is provided, where a suitable dose of noise is added to the abnormal mammograms such that the performance of a suboptimal lesion detector is improved without altering the detector&#39;s parameters. A stochastic resonance noise-based detection approach is presented to improve suboptimal detectors which suffer from model mismatch due to the Gaussian assumption. Furthermore, a stochastic resonance noise-based detection enhancement framework is presented to deal with more general model mismatch cases.

CROSS-REFERENCE TO RELATED APPLICATIONS

This application claims priority to U.S. application Ser. No.11/551,473, filed on Oct. 20, 2006, which claims priority to U.S.Provisional Patent Application No. 60/728,504, filed on Oct. 20, 2005,each of which are incorporated in their respective entireties.

STATEMENT REGARDING FEDERALLY SPONSORED RESEARCH OR DEVELOPMENT

This invention was made with government support under Grant Nos.FA9550-05-C-0139 and FA9550-09-1-0064, each of which was awarded by theAir Force Office of Scientific Research (AFOSR). The government hascertain rights in the invention.

BACKGROUND OF THE INVENTION

1. Field of the Invention

The present invention relates to signal detection and, moreparticularly, to a method for detecting micro-calcifications inmammograms using novel algorithms and stochastic resonance noise.

2. Description of the Related Art

Stochastic resonance (SR) is a nonlinear physical phenomenon in whichthe output signals of some nonlinear systems can be enhanced by addingsuitable noise under certain conditions. The classic SR signature is thesignal-to-noise ratio (SNR) gain of certain nonlinear systems, i.e., theoutput SNR is higher than the input SNR when an appropriate amount ofnoise is added.

Although SNR is a very important measure of system performance, SNRgain-based SR approaches have several limitations. First, the definitionof SNR is not uniform and it varies from one application to another.Second, to optimize the performance, the complete a priori knowledge ofthe signal is required. Finally, for detection problems where the noiseis non-Gaussian, SNR is not always directly related to detectionperformance; i.e., optimizing output SNR does not guarantee optimizingprobability of detection.

In signal detection theory, SR also plays a very important role inimproving the signal detectability. For example, improvement ofdetection performance of a weak sinusoid signal has been reported. Todetect a DC signal in a Gaussian mixture noise background, performanceof the sign detector can be enhanced by adding some white Gaussian noiseunder certain circumstances. For the suboptimal detector known as thelocally optimal detector (LOD), detection performance is optimum whenthe noise parameters and detector parameters are matched. The stochasticresonance phenomenon in quantizers results in a better detectionperformance can be achieved by a proper choice of the quantizerthresholds. Detection performance can be further improved by using anoptimal detector on the output signal. Despite the progress achieved bythe above approaches, the use the SR effect in signal detection systemsis rather limited and does not fully consider the underlying theory ofSR.

Simple and robust suboptimal detectors are used in numerousapplications. To improve a suboptimal detector detection performance,two approaches are widely used. In the first approach, the detectorparameters are varied. Alternatively, when the detector itself cannot bealtered or the optimum parameter values are difficult to obtain,adjusting the observed data becomes a viable approach. Adding adependent noise is not always possible because pertinent priorinformation is usually not available.

For some suboptimal detectors, detection performance can be improved byadding an independent noise to the data under certain conditions. For agiven type of SR noise, the optimal amount of noise can be determinedthat maximizes the detection performance for a given suboptimaldetector. However, despite the progress made, the underlying mechanismof the SR phenomenon as it relates to detection problems has not fullybeen explored. For example, until now the “best” noise to be added inorder to achieve the best achievable detection performance for thesuboptimal detector was not known. Additionally, the optimal level ofnoise that should be used for enhanced performance was also unknown.

Breast cancer is a serious disease with high occurrence rate in women.There is clear evidence which shows that early diagnosis and treatmentof breast cancer can significantly increase the chance of survival forpatients. One of the important early symptoms of breast cancer in themammograms is the appearance of micro-calcification clusters. Anaccurate detection of micro-calcifications is highly desirable to ensureearly diagnosis of breast cancer.

Automatic micro-calcification detection techniques play an importantrole in cancer diagnosis and treatment. This, however, still remains achallenging task.

For example, computer-aided diagnosis (CAD) improves the diagnosticperformance of radiologists and is an effective method for earlydiagnosis thereby increasing survival time for women with breast cancer.While advances have been made in the area of CAD for digital mammograms,the main challenge of accurately identifying breast cancer in digitalmammograms still remains, which is due to the small sizes and subtlecontrast of the lesions compared with the surrounding normal breasttissues.

Much effort has been made for detecting micro-calcifications by usingCAD techniques. Some methods tried to detect micro-calcificationsthrough a modeling procedure. For example, Bazzani et al. and Gurcan etal. detected the micro-calcifications by using Gaussianity tests in thedifference and filtered mammograms, respectively. See Armando Bazzani etal., “Automatic detection of clustered micro-calcifications in digitalmammograms using an SVM classifier,” in Proc. of European Symposium onArtificial Neural Networks Plastics, Bruges, 26-28, April, 2000; M. NafiGurcan, Yasemin Yardimci, and A. Enis Getin, “Influence function basedgaussianity tests for detection of micro-calcifications in mammogramimages,” in Proc. International Conference on Image Processing (ICIP),vol. 3, pp. 407-411, 1999. Karssemeijer modeled the mammograms usingMarkov random fields. See N. Karssemeijer, “Adaptive noise equalizationand recognition of micro-calcification clusters in mammograms,” Int. J.Pattern Recognit. Artificial Intell., vol. 7, no. 6, pp. 1357-1376,1993. Nakayama et al. used a Gaussian probability density function (PDF)to model the abnormal regions in the subband mammograms generated by anovel filter bank. See Ryohei Nakayama et al, “Computer-aided diagnosisscheme using a filter bank for detection of micro-calcification clustersin mammograms,” IEEE Trans. on Biomedical Engineering, vol. 53, no. 2,pp. 273-283, February 2006. Regentova et al. considered the PDFs of themagnitudes of the wavelet coefficients, which are assumed to correspondto two hidden Markov states, to obey zero mean Gaussian distributionswith different variances. See Emma Regentova et al, “Detectingmicro-calcifications in digital mammograms using wavelet domain hiddenmarkov tree model,” in Proc. 28th Annual International Conference of theIEEE Engineering in Medicine and Biology Society 2006 (EMBS '06), pp.1972-1975, 30, Aug.-3, September 2006.

Deepa and Tessamma used the deterministic fractal model to characterizebreast background tissues. See Sankar Deepa and Thomas Tessamma,“Fractal modeling of mammograms based on mean and variance for thedetection of micro-calcifications,” in Proc. International Conference onComputational Intelligence and Multimedia Applications, vol. 2, pp.334-348, 13-15, December, 2007. The challenge for these model-basedmethods is that an accurate model is generally not easy to obtain andmodel mismatch is hard to avoid, so the detection results aredeteriorated. There are also some methods that attempt to avoid thenecessity of modeling during the detection process. For example, in Weiet al., relevance vector machine (RVM) was employed as amicro-calcification classifier, and its parameters were determinedthrough a supervised learning procedure. See Liyang Wei et al.,“Relevance vector machine for automatic detection of clusteredmicro-calcifications,” IEEE Trans. on Medical Imaging, vol. 24, no. 10,pp. 1278-1285, October 2005. Catanzariti et al. trained a three-layerfeed-forward artificial neural network (ANN) to detectmicro-calcifications using the features extracted by a bank of Gaborfilters. See Catanzariti et al, “A CAD system for the detection ofmammographyc micro-calcifications based on Gabor Transform,” in Proc.Nuclear Science Symposium Conference Record, vol. 6, pp. 3599-3603,16-22 Oct. 2004.

Strickland et al., Lemaur et al. and Li and Dong proposed the waveletdomain thresholding techniques to obtain the information of interest forthe detection of micro-calcifications. See R. N. Strickland, “Wavelettransform methods for objects detection and recovery,” IEEE Trans. ImageProcessing, vol. 6, pp. 724-735, May, 1997; G. Lemaur, K. Drouiche, andJ. DeConinck, “Highly regular wavelets for the detection of clusteredmicro-calcifications in mammograms,” IEEE Trans. on Medical Imaging,vol. 22, no. 3, March, 2003; Kai-yang Li and Zheng Dong, “A novel methodof detecting calcifications from mammogram images based on wavelet andsobel detector,” in Proc. 2006 IEEE International Conference onMechatronics and Automation, pp. 1503-1508, June 2006. These methodspartially bypassed the modeling problems, but determination of theoptimum parameters, such as the threshold, is still a very challengingtask, and the detection performance was often affected by the suboptimumparameters. Basically, lesion detection can be considered as an anomalydetection problem. Performance of the detectors is heavily dependent onthe accuracy of the mathematical models and the detector parameters.However, appropriate models and optimum parameter values are generallyvery difficult to obtain in practical applications, which often resultsin unsatisfactory detection performance in terms of high probability offalse alarm (P_(F)) and low probability of detection (P_(D)).

Description of the Related Art Section Disclaimer: To the extent thatspecific publications are discussed above in this Description of theRelated Art Section, or elsewhere herein, these discussions should notbe taken as an admission that the discussed publications (for example,technical/scientific publications) are prior art for patent lawpurposes. For example, some or all of the discussed publications may notbe sufficiently early in time, may not reflect subject matter developedearly enough in time and/or may not be sufficiently enabling so as toamount to prior art for patent law purposes. To the extent that specificpublications are discussed above in this Description of the Related ArtSection, or elsewhere herein, they are all hereby incorporated byreference into this document in their respective entirety(ies).

BRIEF SUMMARY OF THE INVENTION

It is therefore a principal object and advantage of the presentinvention to provide a method for determining the best noise to add toimprove detection of a suboptimal, non-linear detector.

It is an additional object and advantage of the present invention toprovide a method for determining the optimal level of noise for improveddetection.

In accordance with the foregoing objects and advantages, the presentinvention provides a method for signal detection in observed sensor datafor a broad range of electromagnetic or acoustic applications such asradar, sonar, as well as imagery such as visual, hyperspectral, andmulti-spectral. The method of the present invention is applicable inapplications involving non-linear processing of the data. Specifically,the method of the present invention determines the stochastic resonancenoise probability density function to be added to either the observeddata process to optimize detection with no increase in the false alarmrate, or to an image to optimize the detection of signal objects fromthe background. In addition, the method of the present inventiondetermines the conditions required for performance improvement usingadditive stochastic resonance noise. The method of the present inventionalso yields a constant false alarm rate (CFAR) receiver implementation,which is essential in operational conditions in which it is imperativeto maintain false alarm rates without adjusting the detector thresholdlevel.

In accordance with an additional embodiment of the present invention, anapparatus and method for detecting micro-calcifications in mammogramsusing novel algorithms and stochastic resonance noise is provided, wherea suitable dose of noise is added to the abnormal mammograms such thatthe performance of a suboptimal lesion detector is improved withoutaltering the detector's parameters. As discussed further below in theDetailed Description of the Invention section, a stochastic resonancenoise-based detection approach is presented to improve some suboptimaldetectors which suffer from model mismatch due to the Gaussianassumption. Furthermore, a stochastic resonance noise-based detectionenhancement framework is presented to deal with more general modelmismatch cases. The algorithms and the framework are tested on a set of75 representative abnormal mammograms. The results show that they yieldsuperior performance when compared with several other classification anddetection approaches.

BRIEF DESCRIPTION OF THE DRAWINGS

The present invention will be more fully understood and appreciated byreading the following Detailed Description in conjunction with theaccompanying drawings, in which:

FIG. 1 is a graph of the effect of additive noise according to thepresent invention.

FIG. 2 is a graph of the values of F₁ and F₀ as a function of xaccording to the present invention.

FIG. 3 is a graph of the relationship between f₁ and f₀ according to thepresent invention.

FIG. 4 is a graph of the relationship between G(f₀; k), f₀, f_(0i)(k),v_(i)(k) with i=1, 2 and different k value 0, 1 and 2 according to thepresent invention.

FIG. 5 is a graph illustrating different H(x) curves where μ=3, A=1according to the present invention.

FIG. 6 is a graph of P_(D) ^(y) as a function of signal level A inGaussian mixture noise when μ=3 and σ₀=1 according to the presentinvention.

FIG. 7 is a graph of P_(D) ^(y) as a function of σ₀ for different typesof noise enhanced detectors when μ=3 and A=1 according to the presentinvention.

FIG. 8 is a graph of P_(D) ^(y) as a function of μ for different typesof noise enhanced detectors when σ₀=1 and A=1 according to the presentinvention.

FIG. 9 is a graph of the ROC curves for different SR noise enhanced signdetectors when N=30 according to the present invention.

FIG. 10 is a schematic of an SR detection system according to thepresent invention.

FIG. 11 is a schematic of an SR detection system according to thepresent invention.

FIG. 12 is a schematic of APD plots of the real-world mammogrambackground data, simulated Gaussian distribution and heavy tailed SαSdistribution data on a log-log scale, in accordance with an embodimentof the present invention.

FIG. 13 shows the detection results for abnormal mammogram type 1 usingnovel algorithms in accordance with an embodiment of the presentinvention.

FIG. 14 shows the detection results for abnormal mammogram type 2 usingnovel algorithms in accordance with an embodiment of the presentinvention.

FIG. 15 shows the detection results for abnormal mammogram type 3 usingnovel algorithms in accordance with an embodiment of the presentinvention.

DETAILED DESCRIPTION OF THE INVENTION

Referring now to the drawings, wherein like reference numerals refer tolike parts throughout, there is seen in FIG. 1 a chart illustrating theeffective of additive noise on a given signal.

The following definitions serve to clarify the present invention:

The term “constant false alarm rate” (CFAR) refers to the attribute of areceiver that maintains the false alarm rate fixed in the presence ofchanging interference levels.

The term “false alarm” refers to the decision that a signal is presentwhen in fact it is not.

The term “false alarm rate” refers to the rate at which a false alarmoccurs.

The term “fixed detector” refers to a detector comprised of a fixed teststatistic and a fixed threshold.

The term “receiver operating characteristic” (ROC) refers to a plot ofthe probability of detection as a function of the probability of falsealarm for a given detector.

To enhance the detection performance, noise is added to an original dataprocess x to obtain a new data process y given by y=x+n, where n iseither an independent random process with pdf p_(n)(•) or a nonrandomsignal. There is no constraint on n. For example, n can be white noise,colored noise, or even be a deterministic signal A, corresponding top_(n)(n)=δ(n−A). As described herein, depending on the detectionproblem, an improvement of detection performance may not always bepossible. In that case, the optimal noise is equal to zero. The pdf of yis expressed by the convolutions of the pdfs such thatp _(y)(y)=p _(x)(x)*p _(n)(x)=∫_(R) _(N) p _(x)(x)p _(n)(y−x)dx)  (7)

The binary hypotheses testing problem for this new observed data y canbe expressed as:

$\begin{matrix}\left\{ \begin{matrix}{{H_{0}:{p_{y}\left( {y;H_{0}} \right)}} = {\int_{R^{N}}{{p_{0}(x)}{p_{n}\left( {y - x} \right)}\ {\mathbb{d}x}}}} \\{{H_{1}:{p_{y}\left( {y;H_{1}} \right)}} = {\int_{R^{N}}{{p_{1}(x)}{p_{n}\left( {y - x} \right)}\ {\mathbb{d}x}}}}\end{matrix} \right. & (8)\end{matrix}$

Since the detector is fixed, i.e., the critical function φ of y is thesame as that for x, the probability of detection based on data y isgiven by,

$\begin{matrix}\begin{matrix}{P_{D}^{y} = {\int_{R^{N}}{{\phi(y)}{p_{y}\left( {y;H_{1}} \right)}\ {\mathbb{d}y}}}} \\{= {\int_{R^{N}}{{\phi(y)}{\int_{R^{N}}{{p_{1}(x)}{p_{n}\left( {y - x} \right)}\ {\mathbb{d}x}\ {\mathbb{d}y}}}}}} \\{= {\int_{R^{N}}{{p_{1}(x)}\left( {\int_{R^{N}}{{\phi(y)}{p_{n}\left( {y - x} \right)}\ {\mathbb{d}y}}} \right)\ {\mathbb{d}x}}}} \\\left. {= {{\int_{R^{N}}{{p_{1}(x)}{C_{n,\phi}(x)}\ {\mathbb{d}x}}} = {E_{1}\left\lbrack {C_{n,\phi}(x)} \right\rbrack}}} \right)\end{matrix} & (9)\end{matrix}$WhereC _(n,φ)(x)≡∫_(R) _(N) φ(y)p _(n)(y−x)dy  (10)

Alternatively,

$\begin{matrix}\begin{matrix}{P_{D}^{y} = {\int_{R^{N}}{{p_{n}(x)}\left( {\int_{R^{N}}{{\phi(y)}{p_{1}\left( {y - x} \right)}\ {\mathbb{d}y}}} \right)\ {\mathbb{d}x}}}} \\{= {\int{{F_{1,\phi}(x)}{p_{n}(x)}\ {\mathbb{d}x}}}} \\{= {E_{n}\left( {F_{1,\phi}(x)} \right)}}\end{matrix} & (11)\end{matrix}$

Similarly,

$\begin{matrix}\begin{matrix}{P_{FA}^{y} = {{\int_{R^{N}}{{p_{0}(x)}{C_{n,\phi}(x)}\ {\mathbb{d}x}}} = {E_{0}\left\lbrack {C_{n,\phi}(x)} \right\rbrack}}} \\{= {{\int{{F_{0,\phi}(x)}{p_{n}(x)}{\mathbb{d}x}}} = {E_{n}\left( {F_{0,\phi}(x)} \right)}}}\end{matrix} & \begin{matrix}\begin{matrix}(12) \\\;\end{matrix} \\(13)\end{matrix}\end{matrix}$whereF _(i,φ)(x)≡∫_(R) _(N) φ(y)p _(i)(y−x)dy i=0,1,  (14)corresponding to hypothesis H_(i). E_(i)(•), E_(n)(•) are the expectedvalues based on distributions p_(i) and p_(n), respectively, and P^(x)_(FA)=F_(0,φ)(0), P^(x) _(D)=F_(1,φ)(0). To simplify notation, subscriptφ of F and C may be omitted and denotes as F₁, F₀, and C_(n),respectively. Further, F₁(x₀) and F₀(x₀) are actually the probability ofdetection and probability of false alarm, respectively, for thisdetection scheme with input y=x+x₀. For example, F₁(−2) is the P_(D) ofthis detection scheme with input x−2. Therefore, it is very convenientto obtain the F₁ and F₀ values by analytical computation if p₀, p₁ and φare known. When they are not available, F₁ and F₀ can be obtained fromthe data itself by processing it through the detector and recording thedetection performance. The optimal SR noise definition may be formalizedas follows.

Consider the two hypotheses detection problem. The pdf of optimum SRnoise is given by

$\begin{matrix}{p_{n}^{opt} = {\arg\;{\max\limits_{p_{n}}{\int_{R^{N}}{{F_{1}(x)}{p_{n}(x)}\ {\mathbb{d}x}}}}}} & (15)\end{matrix}$where

-   -   1) p_(n)(x)≧0, x∈R^(N).    -   2) ∫_(R) _(N) p_(n)(x)dx=1.    -   3) ∫_(R) _(N) F₀(x)p_(n)(x)dx≦F₀(0).

Conditions 1) and 2) are fundamental properties of a pdf function.Condition 3) ensures that P^(y) _(FA)≦P^(x) _(FA), i.e., the P_(FA)constraint specified under the Neyman-Pearson Criterion is satisfied.Further, if the inequality of condition 3) becomes equality, theConstant False Alarm Rate (CFAR) property of the original detector ismaintained. A simple illustration of the effect of additive noise isshown in FIG. 1. In the example,

${{F_{1}\left( {- A} \right)} = {{\max\limits_{x}{{F_{1}(x)}\mspace{14mu}{and}\mspace{14mu}{F_{0}\left( {- A} \right)}}} < {F_{0}(0)}}},$hencep _(n) ^(opt)=δ(x+A)which means the optimal SR noise n=−A is a dc signal with value −A. Inpractical applications, some additional restrictions on the noise mayalso be applied. For example, the type of noise may be restricted,(e.g., n may be specified as Gaussian noise), or we may require a noisewith even symmetric pdf p_(n)(x)=p_(n)(−x) to ensure that the mean valueof y is equal to the mean value of x. However, regardless of theadditional restrictions, the conditions 1), 2), and 3) are always validand the optimum noise pdf can be determined for these conditions.

In general, for optimum SR noise detection in Neyman-Peason detection,it is difficult to find the exact form of p_(n)(•) directly because ofcondition 3). However, an alternative approach considers therelationship between p_(n)(x) and F_(i)(x). From equation (14), for agiven value f₀ of F₀, we have x=F₀ ⁻¹(f₀), where F₀ ⁻¹ is the inversefunction of F₀. When F₀ is a one-to-one mapping function, x is a uniquevector. Otherwise, F₀ ⁻¹(f₀) is a set of x for which F₀(x)=f₀.Therefore, we can express a value or a set of values f₁ of F₁ asf ₁ =F ₁(x)=F ₁(F ₀ ⁻¹(f ₀))  (16)

Given the noise distribution of p_(n)(•) in the original RN domain,p_(n,f) ₀ (•), the noise distribution in the f₀ domain can also beuniquely determined. Further, the conditions on the optimum noise can berewritten in terms of f₀ equivalently as

-   -   4) p_(n,f) ₀ (f₀)≧0    -   5) ∫p_(n,f) ₀ (f₀)df₀=1    -   6) ∫f₀p_(n,f) ₀ (f₀)df₀≦P_(FA) ^(x)        and        P _(D) ^(y)=∫₀ ¹ f ₁ p _(n,f) ₀ (f ₀)df ₀,  (17)        where p_(n,f) ₀ (•) is the SR noise pdf in the f₀ domain.

Compared to the original conditions 1), 2) and 3), this equivalent formhas some advantages. First, the problem complexity is dramaticallyreduced. Instead of searching for an optimal solution in R^(N), thepresent invention seeks an optimal solution in a single dimensionalspace. Second, by applying these new conditions, the present inventionavoids the direct use of the underlying pdfs p₁(•) and p_(o)(•) andreplace them with f₁ and f₀. Note that, in some cases, it is not veryeasy to find the exact form of f₀ and f₁. However, recall that F₁(x₀)and F₀(x₀) are the Probability of Detection and Probability of FalseAlarm, respectively, of the original system with input x+x_(o). Inpractical applications, the relationship may be determined by MonteCarlo simulation using importance sampling. In general, compared to p₁and p_(o), f₁ and f₀ are much easier to estimate and once the optimump_(n,f) ₀ is found, the optimum p_(n)(x) is determined as well by theinverse of the functions F₀ and F₁.

Consider the function J(t), such that J(t)=sup(f₁: f₀=t) is the maximumvalue of f₁ given f₀. Clearly, J(P^(x) _(FA))≧F₁(0)=P^(x) _(D). Itfollows that for any noise p_(n),P _(D) ^(y)(p _(n))=∫₀ ¹ J(f ₀)p _(n,f) ₀ (f ₀)df ₀  (18)Therefore, the optimum P_(D) ^(y) is attained when f₁(f₀)=J(f₀) andP_(D,opt) ^(y)=E_(n)(J).

Improvability of a given detector when SR noise is added can bedetermined by computing and comparing P_(D,opt) ^(y) and P_(D) ^(x).When P_(D,opt) ^(y)>P_(D) ^(x), the given detector is improvable byadding SR noise. However, it requires the complete knowledge of F₁(•)and F₀(•) and significant computation. For a large class of detectors,however, depending on the specific properties of J, it is possible todetermine the sufficient conditions for improvability andnon-improvability more easily. The conditions are determined using thefollowing theorems.

Theorem 1 (Improvability of Detection via SR): If J(P_(FA) ^(x))>P_(D)^(x) or J″(P_(FA) ^(x))>0 when J(t) is second order continuouslydifferentiable around P_(FA) ^(x), then there exists at least one noiseprocess n with pdf p_(n)(•) that can improve the detection performance.

Proof: First, when J(P_(FA) ^(x))>P_(D) ^(x), from the definition of Jfunction, we know that there exists at one least one n₀ such thatF₀(n₀)=P_(FA) ^(x) and F₁(n₀)=J(P_(FA) ^(x))>P_(D) ^(x). Therefore, thedetection performance can be improved by choosing a SR noise pdfP_(n)(n)=δ(n−n₀). When J″(P_(FA) ^(x))>0 and is continuous around P_(FA)^(x), there exists an ∈>0 such that J″(•)>0 on I=(P_(FA) ^(x)−∈, P_(FA)^(x)+∈). Therefore, from Theorem A-1, J is convex on I. Next, add noisen with pdf p_(n)(x)=½ δ(x−x₀)+½δ(x+x₀), where F₀(x₀)=P_(FA) ^(x)+∈/2 andF₀(x₁)=P_(FA) ^(x)−∈/2. Due to the convexity of J,

$P_{D}^{y} = {\frac{{J\left( {P_{FA}^{x} - \frac{ɛ}{2\;}} \right)} + {J\left( {P_{FA}^{x} - \frac{ɛ}{2}} \right)}}{2} > {J\left( P_{FA}^{x} \right)} \geq P_{D}^{x}}$Thus, detection performance can be improved via the addition of SRnoise.

Theorem 2 (Non-improvability of Detection via SR): If there exists anon-decreasing concave function Ψ(f₀) where Ψ(P^(x) _(FA))=J(P^(x)_(FA))=F₁(f₀) and Ψ(f₀)≧J(f₀) for every f₀, then P_(D) ^(y)≦P_(D) ^(x)for any independent noise, i.e., the detection performance cannot beimproved by adding noise.

Proof: For any noise n and corresponding y, we have

$\begin{matrix}{{P_{D}^{y}\left( p_{n} \right)} = {{{\int_{0}^{1}{{J\left( f_{0} \right)}{p_{n,f_{0}}\left( f_{0} \right)}\ {\mathbb{d}f_{0}}}} \leq {\int_{0}^{1}{{\Psi\left( f_{0} \right)}{p_{{nf}_{0}}\left( f_{0} \right)}\ {\mathbb{d}f_{0}}}} \leq {\Psi\left( {\int_{0}^{1}{f_{0}{p_{{nf}_{0}}\left( f_{0} \right)}\ {\mathbb{d}f_{0}}}} \right)} \leq {\Psi\left( P_{FA}^{x} \right)}} = P_{D}^{x}}} & (19)\end{matrix}$The third inequality of the Right Hand Side (RHS) of (19) is obtainedusing the concavity of the Ψ function. The detection performance cannotbe improved via the addition of SR noise.

Before determining the form of the optimum SR noise PDF, i.e., the exactpdf of p_(n) ^(opt), the following result for the form of optimum SRnoise must be determined.

Theorem 3 (Form of Optimum SR Noise): To maximize P_(D) ^(y), under theconstraint that P_(FA) ^(y)≦P_(FA) ^(x), the optimum noise can beexpressed as:p _(n) ^(opt)(n)=λδ(n−n ₁)+(1−λ)δ(n−n ₂)  (20)where 0≦λ≦1. In other words, to obtain the maximum achievable detectionperformance given the false alarm constraints, the optimum noise is arandomization of two discrete vectors added with the probability λ and1−λ, respectively.

Proof: Let U={(f₁, f₀)|f₁=F₁(x), f₀=F₀(x), x

R^(N)) be the set of all pairs of (f₁; f₀). Since 0≦f₁; f₀≦1, U is asubset of the linear space R². Furthermore, let V be the convex hull ofU. Since V⊂R², its dimension Dim(V)≦2. Similarly, let the set of allpossible (P_(D) ^(y); P_(FA) ^(y)) be W. Since any convex combination ofthe elements of U, say

$\left( {\chi,\phi} \right) = {\sum\limits_{i = 1}^{M}\;{\alpha_{i}\left( {f_{1,i},f_{0,i}} \right)}}$can be obtained by setting the SR noise pdf such that

${p_{n,f_{o}}\left( f_{0} \right)} = {\sum\limits_{i = 1}^{M}{\alpha_{i}{\delta\left( {f_{0} - f_{0,i}} \right)}}}$V⊂W. It can also be shown that W⊂V. Otherwise, there would exist atleast one element z such that z

W, but z∉V. In this case, there exists a small set S and a positivenumber τ such thatS={(x,y)|∥(x,y)−z∥ ₂ ²<τ} and S∩V=‘{ }’where ‘{ }’ denotes an empty set. However, since 0≦f₁; f₀≦1, by the wellknown property of integration, there always exists a finite set E withfinite elements such that E⊂U and (x₁; y₁), a convex combination of theelements of E, such that∥(x ₁ ,y ₁)−z∥ ₂ ²<τ

Since (x₁; y₁)

V, then (x₁; y₁)

(V∩S) which contradicts the definition of S. Therefore, W⊂V. Hence, W=V.From Theorem A-4, (P^(y) _(D); P^(y) _(FA)) can be expressed as a convexcombination of three elements. Also, since we are only interested inmaximizing P_(D) under the constraint that P_(FA) ^(y)≦P_(FA) ^(x), theoptimum pair can only belong to B, the set of the boundary elements ofV. To show this, let (f₁*; f₀*) be an arbitrary non-boundary pointinside V. Since there exists a τ>0 such that (f₁*+τ, f₀*)

V, then (f₁*; f₀*) is inadmissible as an optimum pair. Thus, the optimumpair can only exist on the boundary and each z on the boundary of V canbe expressed as the convex combination of only two elements in U. Hence,(P _(D,opt) ^(y) ,P _(FA,opt) ^(y))=λ(f ₁₁ ,f ₀₁)+(1−λ)(f ₁₂ ,f₀₂)  (21)where (f₁₁; f₀₁); (f₁₂; f₀₂)

U, 0≦λ≦1. Therefore, we havep _(n,f) ₀ ^(opt)=λδ(f ₀ −f ₀₁)+(1−λ)δ(f ₀ −f ₀₂)  (22)

Equivalently, p_(n) ^(opt)(n)=λδ(n−n₁)+(1−λ)δ(n−n₂), where n₁ and n₂ aredetermined by the equations

$\begin{matrix}\left\{ \begin{matrix}{{F_{0}\left( n_{1} \right)} = f_{01}} \\{{F_{1}\left( n_{1} \right)} = f_{11}} \\{{F_{0}\left( n_{2} \right)} = f_{02}} \\{{F_{1}\left( n_{2} \right)} = f_{12}}\end{matrix} \right. & (23)\end{matrix}$

Alternatively, the optimum SR noise can also be expressed in terms ofC_(n), such thatC _(n) ^(opt)(x)=λφ(x+n ₁)+(1−λ)φ(x+n ₂)  (24)

From equation (22), we haveP _(D,opt) ^(y) =λJ(f ₀₁)+(1−λ)J(f ₀₂)  (25)andP _(FA,opt) ^(y) =λf ₀₁+(1−λ)f ₀₂ ≦P _(FA) ^(x)  (26)

Depending on the location of the maxima of J(•), determination of thepdf of optimum SR noise may be accomplished according to the followingtheorem.

Theorem 4: Let

$F_{1M} = {{\max\;\left( {J(t)}\; \right)\mspace{11mu}{and}\mspace{14mu} t_{0}} = {\arg\;{\min\limits_{t}{\left( {{J(t)} = F_{1M}} \right).}}}}$It follows that

Case 1: If t_(o)≦P_(FA) ^(x) then P_(FA,opt) ^(x)=t_(o) and P_(D,opt)^(y)=F_(1M), i.e., the maximum achievable detection performance isobtained when the optimum noise is a DC signal with value no, i.e.,p _(n) ^(opt)(n)=δ(n−n ₀)  (27)where F₀(n_(o))=t_(o) and F₁(n_(o))=F_(1M).

Case 2: If t_(o)>P_(FA) ^(x), then P_(FA,opt) ^(x)=F₀(0)=P_(FA) ^(x),i.e., the inequality of (26) becomes equality. Furthermore,P _(FA,opt) ^(y) =λf ₀₁+(1−λ)f ₀₂ =P _(FA) ^(x)  (28)

Proof: For Case 1, notice thatP _(D) ^(y)=∫₀ ¹ J(f ₀)P _(n,f) ₀ (f ₀)df ₀≦∫₀ ¹ F _(1M) p _(n,f) ₀ (f₀)df ₀ =F _(1M)and F₁(n₀)=F_(1M). Therefore the optimum detection performance isobtained when the noise is a DC signal with value n₀ with P_(FA)^(y)=t₀.

The contradiction method is used to prove Case 2. First, supposing thatthe optimum detection performance is obtained when P_(FA,opt)^(y)=κ<P_(FA) ^(x) with noise pdf p_(n,f) ₀ ^(opt)(f₀). Let

${p_{n_{1},f_{0}}\left( f_{0} \right)} = {{\frac{P_{FA}^{x} - \kappa}{t_{0} - \kappa}{\delta\left( {f_{0} - t_{0}} \right)}} + {\frac{t_{0} - P_{FA}^{x}}{t_{0} - \kappa}{{p_{n,f_{0}}^{opt}\left( f_{0} \right)}.}}}$It is easy to verify that p_(n,fo)(f₀) is a valid pdf. Let y₁=x+n1. Wenow have

${P_{FA}^{y_{1}} = {{{\frac{P_{FA}^{x} - \kappa}{t_{0} - \kappa}t_{0}} + {\frac{t_{0} - P_{FA}^{x}}{t_{0} - \kappa}\kappa}} = P_{FA}^{x}}},{and}$$P_{D}^{y_{1}} = {{{\frac{P_{FA}^{x} - \kappa}{t_{0} - \kappa}F_{1M}} + {\frac{t_{0} - P_{FA}^{x}}{t_{0} - \kappa}P_{D,{opt}}^{y}}} > P_{D,{opt}}^{y}}$

But this contradicts (15), the definition of p_(n) ^(opt). Therefore,P_(FA,opt) ^(y)=P_(FA) ^(x), i.e., the maximum achievable detectionperformance is obtained when the probability of false alarm remains thesame for the SR noise modified observation y.

For Case 2 of Theorem 4, i.e., when t₀>P_(FA) ^(x), let us consider thefollowing construction to derive the form of the optimum noise pdf. FromTheorem 4, we have the condition that P_(FA,opt) ^(y)=F₀(0)=P_(FA) ^(x)is a constant. Define an auxiliary function G such thatG(f ₀ ,k)=J(f ₀)−kf ₀,  (29)where k

R. We have P_(D) ^(y)=E_(n)(J)=E_(n)(G(f₀, k))+kE_(n)(f₀)=E_(n)(G(f₀;k))+k P_(FA) ^(x). Hence, p_(n,f) ₀ ^(opt) also maximizes E_(n)(G(f₀;k)) and vice versa. Therefore, under the condition that P_(FA)^(y)=P_(FA) ^(x), maximization of P_(D) ^(y) is equivalent tomaximization of E_(n)(G(f₀; k)). Divide the domain of f₀ into twointervals I₁=[0, P_(FA) ^(x)] and I₂=[P_(FA) ^(x), 1]. Let f₀₁(k) be theminimum value that maximizes G(f₀; k) in I₁ and let f₀₂(k) be theminimum value that maximizes G(f₀; k) in I₂. Also, let v₁(k)=G(f₀₁; k)and v₂(k)=G(f₀₂; k) be the corresponding maximum values. Since for anyf₀, G(f₀; k) is monotonically decreasing when k is increasing, v₁(k) andv₂(k) are monotonically decreasing while f₀₁(k) and f₀₂(k) aremonotonically non-increasing when k is increasing. Since G(f₀; 0)=J,therefore v₂(0)=F_(1M)>v₁(0), furthermore, when k is very large, we havev₁(k)=J(0)>v₂(k)=J(P_(FA) ^(x))−k P_(FA) ^(x). Hence, there exists atleast one k₀>0 such that v₁(k₀)=v₂(k₀)≡v. For illustration purposes, theplots of G(f₀; k) for the detection problem discussed below are shown inFIG. 4. Divide the [0,1] interval into two non-overlapping parts A,{f₀₁(k₀), f₀₂(k₀)}, such that {f₀₁(k₀); f₀₂(k₀)}∪A=[0, 1] and {f₀₁(k₀);f₀₂(k₀)}∩A={ }. Next, represent p_(n,fo)(f₀) asp _(n,f) ₀ (f ₀)=α₁δ(f ₀ −f ₀₁(k ₀))+α₂δ(f ₀ −f ₀₂(k ₀))+I _(A)(f ₀)p_(n,f) ₀ (f ₀)  (30)where I_(A)(f₀)=1 for f₀

A and is zero otherwise (an indicator function). From equation (5), wemust have

$\begin{matrix}{\mspace{40mu}{{{\alpha_{1} + \alpha_{2} + {\int_{A}{p_{n,f_{0}}{\mathbb{d}f_{0}}}}} = 1},\;{and}}\mspace{281mu}} & (31) \\{{E_{n}(G)} = {{{\left( {\alpha_{1} + \alpha_{2}} \right)v} + {\int_{A}{{G\left( {f_{0},k_{0}} \right)}{pn}_{,f_{0}}{\mathbb{d}f_{0}}}}} = {{v + {\int_{A}{\underset{\underset{\leq 0}{︸}}{\left( {{G\left( {f_{0},k_{0}} \right)} - v} \right)}{pn}_{,f_{0}}{\mathbb{d}f_{0}}}}} \leq v}}} & (32)\end{matrix}$

Note that J(f₀)≦v for all f₀

A. Clearly, the upper bound can be attained when p_(n,f) ₀ =0 for all f₀

A, i.e., α₁+α₂=1. Therefore, P_(D,opt) ^(y) P_(D,opt) ^(y)=E_(n)(G)+k₀P_(FA) ^(x)=v+k₀ P_(FA) ^(x). From equation (28), we have

$\begin{matrix}{{p_{n,f_{0}}^{opt}\left( f_{0} \right)} = {{\frac{{f_{02}\left( k_{0} \right)} - P_{FA}^{x}}{{f_{02}\left( k_{0} \right)} - {f_{01}\left( k_{0} \right)}}{\delta\left( {f_{0} - {f_{01}\left( k_{0} \right)}} \right)}} + {\frac{P_{FA}^{x} - {f_{01}\left( k_{0} \right)}}{{f_{02}\left( k_{0} \right)} - {f_{01}\left( k_{0} \right)}}{\delta\left( {f_{0} - {f_{02}\left( k_{0} \right)}} \right)}}}} & (33)\end{matrix}$

Notice that by letting

${\lambda = \frac{{f_{02}\left( k_{0} \right)} - P_{FA}^{x}}{{f_{02}\left( k_{0} \right)} - {f_{01}\left( k_{0} \right)}}},$(33) is equivalent to (22).

Equivalently, we have the expression of p_(n) ^(opt)(n) as

$\begin{matrix}{{p_{n}^{opt}(n)} = {{\frac{{f_{02}\left( k_{0} \right)} - P_{FA}^{x}}{{f_{02}\left( k_{0} \right)} - {f_{01}\left( k_{0} \right)}}{\delta\left( {n - n_{1}} \right)}} + {\frac{P_{FA}^{x} - {f_{01}\left( k_{0} \right)}}{{f_{02}\left( k_{0} \right)} - {f_{01}\left( k_{0} \right)}}{\delta\left( {n - n_{2}} \right)}}}} & (34)\end{matrix}$

Further, in the special case where f₁ is continuously differentiable, Gis also continuously differentiable. Since f₀₁ and f₀₂ are at leastlocal maxima, we have

${\frac{\partial G}{\partial f_{0}}\left( {f_{01},k_{0}} \right)} = {{\frac{\partial G}{\partial f_{0}}\left( {f_{02},k_{0}} \right)} = 0}$

Therefore, from the derivative of (29), we have

$\begin{matrix}{{{\frac{\mathbb{d}J}{\mathbb{d}f_{0}}\left( {f_{01}\left( k_{0} \right)} \right)} = {{\frac{\mathbb{d}J}{\mathbb{d}f_{0}}\left( {f_{02}\left( k_{0} \right)} \right)} = k_{0}}}{{{J\left( {f_{02}\left( k_{0} \right)} \right)} - {J\left( {f_{01}\left( k_{0} \right)} \right)}} = {k_{0}\left( {{f_{02}\left( k_{0} \right)} - \left( {f_{01}\left( k_{0} \right)} \right)} \right.}}} & {(35)\mspace{14mu}{and}\mspace{14mu}(36)}\end{matrix}$

In other words, the line connecting (J(f₀₁(k₀)), f₀₁(k₀)) andJ(f₀₂(k₀)), f₀₂(k₀)) is the bi-tangent line of J(•) and k₀ is its slope.Also,P _(D,opt) ^(y) =v+k ₀ P _(FA) ^(x)  (37)

Thus, the condition under which SR noise can improve detectionperformance has been derived, and the specific form of the optimum SRnoise has been obtained.

Detection Example

In a detection problem where two hypotheses H0 and H1 are given as

$\begin{matrix}\left\{ \begin{matrix}{{H_{0}:{x\lbrack i\rbrack}} = {w\lbrack i\rbrack}} \\{{H_{1}:{x\lbrack i\rbrack}} = {A + {w\lbrack i\rbrack}}}\end{matrix} \right. & (38)\end{matrix}$for i=0, 1, . . . , N−1, A>0 is a known dc signal, and w[i] are i.i.dnoise samples with a symmetric Gaussian mixture noise pdf as follows

$\begin{matrix}{{{p_{w}(w)} = {{\frac{1}{2}{\gamma\left( {{w;{- \mu}},\sigma_{0}^{2}} \right)}} + {\frac{1}{2}{\gamma\left( {{w;\mu},\sigma_{0}^{2}} \right)}}}}{{and}\mspace{14mu}{where}}{{\gamma\left( {{w;\mu},\sigma_{0}^{2}} \right)} = {\frac{1}{\sqrt{2{\pi\sigma}^{2}}}{\exp\left\lbrack {- \frac{\left( {w - \mu} \right)^{2}}{2\sigma^{2}}} \right\rbrack}}}} & (39)\end{matrix}$setting μ=3, A=1 and σ₀=1. A suboptimal detector is considered with teststatistic

$\begin{matrix}{{T(x)} = {\frac{1}{N}{\sum\limits_{i = 0}^{N - 1}\left( {{\frac{1}{2} + {\frac{1}{2}{{sgn}\left( {x\lbrack i\rbrack} \right)}}} = {{\frac{1}{N}{\sum\limits_{i = 0}^{N - 1}{\left( {\varpi_{x}\lbrack i\rbrack} \right){where}{\varpi_{x}\lbrack i\rbrack}}}} = {\frac{1}{2} + {\frac{1}{2}{{{sgn}\left( \lbrack i\rbrack \right)}.}}}}} \right.}}} & (40)\end{matrix}$

From equation (40), this detector is essentially a fusion of thedecision results of N i.i.d. sign detectors.

When N=1, the detection problem reduces to a problem with the teststatistic T₁(x)=x, threshold η=0 (sign detector) and the probability offalse alarm P_(FA) ^(x)=0.5. The distribution of x under the H₀ and H₁hypotheses can be expressed as

$\begin{matrix}{{{p_{0}(x)} = {{\frac{1}{2}{\gamma\left( {{x;{- \mu}},\sigma_{0}^{2}} \right)}} + {\frac{1}{2}{\gamma\left( {{x;\mu},\sigma_{0}^{2}} \right)}}}}{and}} & (41) \\{{p_{1}(x)} = {{\frac{1}{2}{\gamma\left( {{x;{{- \mu} + A}},\sigma_{0}^{2}} \right)}} + {\frac{1}{2}{\gamma\left( {{x;{\mu + A}},\sigma_{0}^{2}} \right)}}}} & (42)\end{matrix}$respectively. The critical function is given by

$\begin{matrix}{{\phi(x)} = \left\{ {\begin{matrix}1 & {x > 0} \\0 & {x \leq 0}\end{matrix}.} \right.} & (43)\end{matrix}$

The problem of determining the optimal SR noise is to find the optimalp(n) where for the new observation y=x+n, the probability of detectionP_(D) ^(y)=p(y>0;H₁) is maximum while the probability of false alarmP_(FA) ^(y)=p(y>0;H₀)≦P_(FA) ^(x)=½.

When N>1, the detector is equivalent to a fusion of N individualdetectors and the detection performance monotonically increases with N.Like the N=1 case, when the decision function is fixed, the optimum SRnoise can be obtained by a similar procedure. Due to space limitations,only the suboptimal case where the additive noise n is assumed to be ani.i.d noise is considered here. Under this constraint, since the P_(D)sand P_(FA)s of each detector are the same, it can be shown that theoptimal noise for the case N>1 is the same as N=1 because P_(FA)≦0.5 isfixed for each individual detector while increasing its P_(D). Hence,only the one sample case (N=1) is considered below. However, theperformance of the N>1 case can be derived similarly.

The determination of the optimal SR noise pdf follows from equations(11) and (13), where it can be shown that in this case,

$\begin{matrix}\begin{matrix}{{F_{1}(x)} = {\int_{0}^{+ \infty}{{\phi(y)}{p_{1}\left( {y - x} \right)}\ {\mathbb{d}y}}}} \\{= {\frac{1}{2}\left( {\int_{0}^{+ \infty}{\left\lbrack {{\gamma\left( {{{y - x};{{- \mu} + A}},\sigma_{0}^{2}} \right)} + {\gamma\left( {{{y - x};{\mu + A}},\sigma_{0}^{2}} \right)}} \right\rbrack{\mathbb{d}y}}} \right)}} \\{= {{\frac{1}{2}{Q\left( \frac{{- x} - \mu - A}{\sigma_{0}} \right)}} + {\frac{1}{2}{Q\left( \frac{{- x} + \mu - A}{\sigma_{0}} \right)}}}}\end{matrix} & (44) \\{\mspace{14mu}{and}\mspace{529mu}} & \; \\\begin{matrix}{{F_{0}(x)} = {\int_{0}^{+ \infty}{{\phi(y)}{p_{0}\left( {y - x} \right)}\ {\mathbb{d}y}}}} \\{= {\frac{1}{2}\left( {\int_{0}^{+ \infty}{\left\lbrack {{\gamma\left( {{{y - x};{- \mu}},\sigma_{0}^{2}} \right)} + {\gamma\left( {{{y - x};\mu},\sigma_{0}^{2}} \right)}} \right\rbrack{\mathbb{d}y}}} \right)}} \\{= {{\frac{1}{2}{Q\left( \frac{{- x} - \mu}{\sigma_{0}} \right)}} + {\frac{1}{2}{Q\left( \frac{{- x} + \mu}{\sigma_{0}} \right)}}}}\end{matrix} & (45) \\{{where}\mspace{495mu}} & \; \\{{{Q(x)} = {\int_{x}^{\infty}{\frac{1}{\sqrt{2\;\pi}}{\exp\left( {{- t^{2}}/2} \right)}\ {{\mathbb{d}t}.}}}}\mspace{214mu}} & \;\end{matrix}$

It is also easy to show that, in this case, F₁(x)>F₀(x) and both aremonotonically increasing with x. Therefore, J(f₀)=f₁(f₀)=f₁, and U=(f₁;f0) is a single curve. FIG. 2 shows the values of f₁ and f₀ as afunction of x, while the relationship between F₁ and F₀ is shown in FIG.3. V, the convex hull of all possible P_(D) and P_(FA) after n is addedand shown as the light and dark shadowed regions in FIG. 3. Note that asimilar non-concave ROC occurs in distributed detection systems anddependent randomization is employed to improve system performance.Taking the derivative of f₁ w.r.t. f₀, we have

$\begin{matrix}{{\frac{\mathbb{d}\left( f_{1} \right)}{\mathbb{d}\left( f_{0} \right)} = {\frac{\frac{\mathbb{d}\left( f_{1} \right)}{\mathbb{d}(x)}}{\frac{\mathbb{d}\left( f_{0} \right)}{\mathbb{d}(x)}} = \frac{p_{1}\left( {- x} \right)}{p_{0}\left( {- x} \right)}}},{and}} & (46) \\\begin{matrix}{\frac{\mathbb{d}^{2}\left( f_{1} \right)}{\mathbb{d}\left( f_{0}^{2} \right)} = {\frac{1}{p_{0}\left( {- x} \right)}\frac{\mathbb{d}\left( \frac{p_{1}\left( {- x} \right)}{p_{0}\left( {- x} \right)} \right)}{\mathbb{d}x}}} \\{{= \frac{{{- {p_{1}^{\prime}\left( {- x} \right)}}{p_{0}\left( {- x} \right)}} + {{p_{0}^{\prime}\left( {- x} \right)}{p_{1}\left( {- x} \right)}}}{p_{0}^{2}\left( {- x} \right)}},}\end{matrix} & (47)\end{matrix}$where x=F₀ ⁻¹(f₀). Since

$\frac{\mathbb{d}{\gamma\left( {{{y - x};\mu},\sigma^{2}} \right)}}{\mathbb{d}x} = {\frac{\mu - x}{\sigma^{2}}{\gamma\left( {{{y - x};\mu},\sigma^{2}} \right)}}$we have p′₀(−x)|_(x=0) and

$\begin{matrix}\begin{matrix}{{{\frac{\mathbb{d}^{2}\left( f_{1} \right)}{\mathbb{d}\left( f_{0}^{2} \right)}}_{f_{0} = {f_{0}{(0)}}} = \frac{\begin{matrix}{{{- {p_{1}^{\prime}\left( {- x} \right)}}{p_{0}\left( {- x} \right)}} +} \\{{p_{0}^{\prime}\left( {- x} \right)}{p_{1}\left( {- x} \right)}}\end{matrix}}{p_{0}^{3}\left( {- x} \right)}}}_{x = 0} \\{{= \frac{- {p_{1}^{\prime}\left( {- x} \right)}}{p_{0}^{2}\left( {- x} \right)}}}_{x = 0} \\{= \frac{\left( {\mu - A} \right){\exp\left( {- \frac{\left( {\mu - A} \right)^{2}}{2\;\sigma_{0}^{2}}} \right)}}{\sqrt{2\;\pi}\sigma_{0}^{3}{p_{0}^{2}(0)}}} \\{= {\frac{\left( {\mu + A} \right){\exp\left( {- \frac{\left( {\mu + A} \right)^{2}}{2\;\sigma_{0}^{2}}} \right)}}{\sqrt{2\;\pi\;\sigma_{0}^{3}{p_{0}^{2}(0)}}}.}}\end{matrix} & (48)\end{matrix}$

With respect to the improvability of this detector, when A<μ, setting(48) equal to zero and solving the equation for σ₀, we have σ₁, the zeropole of (48)

$\sigma_{1} = {\sqrt{2\frac{\mu\; A}{\ln\left( \frac{\mu + A}{\mu - A} \right)}}.}$

When σ₀<σ₁, then

${\frac{\mathbb{d}^{2}\left( f_{1} \right)}{\mathbb{d}\left( f_{0}^{2} \right)}}_{f_{0} = {F_{0}{(0)}}} > 0$and, in this example, σ₁ ²=8.6562>σ₀ ²=1. From Theorem 1, this detectoris improvable by adding independent SR noise. When

${{{A > \mu},\frac{\mathbb{d}^{2}\left( f_{1} \right)}{\mathbb{d}\left( f_{0}^{2} \right)}}}_{f_{0} = {f_{0}{(0)}}} < 0$the improvability cannot be determined by Theorem 1. However, for thisparticular detector, as discussed below, the detection performance canstill be improved.

The two discrete values as well as the probability of their occurrencemay be determined by solving equations (35) and (36). From equations(44) and (45), the relationship between f₁, f₀ and x, and equation (46),we have

$\begin{matrix}{{\frac{p_{1}\left( {- n_{1}} \right)}{p_{0}\left( {- n_{1}} \right)} = \frac{p_{1}\left( {- n_{2}} \right)}{p_{0}\left( {- n_{2}} \right)}}{\frac{{F_{1}\left( n_{1} \right)} - {F_{1}\left( n_{2} \right)}}{{F_{0}\left( n_{1} \right)} - {F_{0}\left( n_{2} \right)}} = {\frac{p_{1}\left( {- n_{2}} \right)}{p_{0}\left( {- n_{2}} \right)}.}}} & (49)\end{matrix}$

Although it is generally very difficult to solve the above equationanalytically, in this particular detection problem,

${{p_{1}\left( {- \left( {\mu - \frac{A}{2}} \right)} \right)} = {{0.5\;{\gamma\left( {{{- \frac{A}{2}};0},\sigma_{0}^{2}} \right)}} + {0.5\;{\gamma\left( {{{{2\;\mu} + \frac{A}{2}};0},\sigma_{0}^{2}} \right)}}}},{{p_{0}\left( {- \left( {\mu - \frac{A}{2}} \right)} \right)} = {{0.5\;{\gamma\left( {{{- \frac{A}{2}};0},\sigma_{0}^{2}} \right)}} + {0.5\;{\gamma\left( {{{{2\;\mu} - \frac{A}{2}};0},\sigma_{0}^{2}} \right)}}}},{{p_{1}\left( {- \left( {{- \mu} - \frac{A}{2}} \right)} \right)} = {{0.5\;{\gamma\left( {{{- \frac{A}{2}};0},\sigma_{0}^{2}} \right)}} + {0.5\;{\gamma\left( {{{{2\;\mu} - \frac{A}{2}};0},\sigma_{0}^{2}} \right)}}}},{{p_{0}\left( {- \left( {{- \mu} - \frac{A}{2}} \right)} \right)} = {{0.5\;{\gamma\left( {{{- \frac{A}{2}};0},\sigma_{0}^{2}} \right)}} + {0.5\;{\gamma\left( {{{{2\;\mu} + \frac{A}{2}};0},\sigma_{0}^{2}} \right)}}}},{{so}\mspace{14mu}{that}}$${\frac{p_{1}\left( {- \left( {\mu - \frac{A}{2}} \right)} \right)}{p_{0}\left( {- \left( {\mu - \frac{A}{2}} \right)} \right)} \cong 1},{\frac{p_{1}\left( {- \left( {{- \mu} - \frac{A}{2}} \right)} \right)}{p_{0}\left( {- \left( {{- \mu} - \frac{A}{2}} \right)} \right)} \cong 1}$and${{F_{1}\left( \left( {\mu - \frac{A}{2}} \right) \right)} - {F_{1}\left( \left( {{- \mu} - \frac{A}{2}} \right) \right)}} = {{F_{0}\left( \left( {\mu - \frac{A}{2}} \right) \right)} - {F_{0}\left( \left( {{- \mu} - \frac{A}{2}} \right) \right)}}$given 2μ−A/2>3σ₀. Thus, the roots n₁; n₂ of equation (49) can beapproximately expressed as n₁=−μ−A/2 and n₂=μ−A/2.

Correspondingly,

$\lambda = {{{\frac{{F_{0}\left( n_{2} \right)} - {F_{0}(0)}}{{F_{0}\left( n_{2} \right)} - {F_{0}\left( n_{1} \right)}}\mspace{14mu}{and}\mspace{14mu} 1} - \lambda} = {\frac{{F_{0}(0)} - {F_{0}\left( n_{1} \right)}}{{F_{0}\left( n_{2} \right)} - {F_{0}\left( n_{1} \right)}}.}}$

Hence

$\begin{matrix}{\begin{matrix}{{p_{n}^{opt}(n)} = {{\lambda\;{\delta\left( {n - n_{1}} \right)}} + {\left( {1 - \lambda} \right){\delta\left( {n - n_{2}} \right)}}}} \\{{= {{0.3085\;{\delta\left( {n + 3.5} \right)}} + {0.6915\;{\delta\left( {n - 2.5} \right)}}}},}\end{matrix}{and}} & (50) \\{P_{D,{opt}}^{y} = {{{\lambda\;{F_{1}\left( n_{1} \right)}} + {\left( {1 - \lambda} \right){F_{1}\left( n_{2} \right)}}} = {0.6915.}}} & (51)\end{matrix}$

The present invention also encompasses special cases where the SR noiseis constrained to be symmetric. These include symmetric noise witharbitrary pdf p_(s)(x), white Gaussian noise p_(g)(x)=γ(x; 0, σ²) andwhite uniform noise p_(u)(x)=1/a, a>0, −a/2≦x≦a/2. The noise modifieddata processes are denoted as y_(s), y_(g) and y_(u), respectively.Here, for illustration purposes, the pdfs of these suboptimal SR noisesmay be found by using the C(x) functions. The same results can beobtained by applying the same approach as in the previous subsectionusing F₁(•) and F₀(•) functions. For the arbitrary symmetrical noisecase, we have the conditionp _(s)(x)=p _(s)(−x).  (52)

Therefore, p(y|H₀) is also a symmetric function, so that P_(FA) ^(y)^(s) =½. By equations (43) and (52), we have

$\begin{matrix}\begin{matrix}{{C_{s}(x)} = {\int_{0}^{\infty}{{p_{s}\left( {t - x} \right)}\ {\mathbb{d}t}}}} \\{= {\int_{- x}^{\infty}{{p_{s}(t)}\ {\mathbb{d}t}}}} \\{= {\int_{- \infty}^{x}{{p_{s}(t)}\ {\mathbb{d}t}}}} \\{= {1 - {{C_{s}\left( {- x} \right)}.}}}\end{matrix} & (53)\end{matrix}$

Since p_(s)(x)≧0, we also haveC _(s)(x ₁)≧C _(s)(x ₀) for any x ₁ ≧x ₀,  (54) andC _(s)(0)=½, C _(s)(−∞)=0, and C _(s)(∞)=2  (55)

From equations (9) and (53), we have the P_(D) of y_(s) given by

$\begin{matrix}\begin{matrix}{P_{D}^{y_{s}} = {\int_{- \infty}^{\infty}{{p_{1}(x)}{C_{s}(x)}\ {\mathbb{d}x}}}} \\{= {{\int_{- \infty}^{0}{{p_{1}(x)}{C_{s}(x)}\ {\mathbb{d}x}}} + {\int_{0}^{\infty}{\left( {1 - {C_{s}\left( {- x} \right)}} \right){p_{1}(x)}\ {\mathbb{d}x}}}}} \\{= {{\int_{- \infty}^{0}{\left( {{p_{1}(x)} - {p_{1}\left( {- x} \right)}} \right){C_{s}(x)}\ {\mathbb{d}x}}} + P_{D}^{x}}} \\{{= {{\int_{- \infty}^{0}{{H(x)}{C_{s}(x)}\ {\mathbb{d}x}}} + P_{D}^{x}}},}\end{matrix} & (56)\end{matrix}$where H(x)≡p₁(x)−p₁(−x). FIG. 5 shows a plot of H(x) for several σ₀values. Finally, from equation (42), we have

${p_{1}\left( {- x} \right)} = {{\frac{1}{2}{\gamma\left( {{x;{\mu - A}},\sigma_{0}^{2}} \right)}} + {\frac{1}{2}{{\gamma\left( {{x;{{- \mu} - A}},\sigma_{0}^{2}} \right)}.}}}$

When A≧μ, since p₁(−x)≧p₁(x) when x<0, we have, H(x)<0, x<0. Fromequation (56), P_(D) ^(y)≦P_(D) ^(x) for any H(x), i.e., in this case,the detection performance of this detector cannot be improved by addingsymmetric noise. When A<μ and σ₀≧σ₁ then H(x)<0, ∀x<0. Therefore, addingsymmetric noise will not improve the detection performance as well.However, when σ₀<σ₁, H(x) has only a single root x₀ for x<0 and H(x)<0,∀x<x₀, H(x)>0, ∀x

(x₀, 0) and detection performance can be improved by adding symmetric SRnoise. From equation (56), we have

$\begin{matrix}{{C_{s}^{opt}(x)} = \left\{ {{\begin{matrix}{0,} & {x < x_{0}} \\{\frac{1}{2},} & {{x_{0} \leq x \leq 0},}\end{matrix}{and}p_{s}^{opt}} = {{\frac{1}{2}{\delta\left( {x - x_{0}} \right)}} + {\frac{1}{2}{{\delta\left( {x + x_{0}} \right)}.}}}} \right.} & (57)\end{matrix}$

Furthermore, since γ(−μ; −μ−A,σ² ₀)=γ(−μ; −μ+A,σ² ₀) and γ(−μ; −μ+A,σ²₀)≈0 given 2μ−A>>σ₀, we have x₀≈−μ. Therefore,

$\begin{matrix}{p_{s}^{opt} = {{\frac{1}{2}{\delta\left( {x - \mu} \right)}} + {\frac{1}{2}{{\delta\left( {x + \mu} \right)}.}}}} & (58)\end{matrix}$

The pdf of y for the H₁ hypothesis becomes

$\begin{matrix}{{p_{1,y_{s}}^{opt}(y)} = {{\frac{1}{2}{\gamma\left( {{y;A},\sigma_{0}^{2}} \right)}} + {\frac{1}{4}{\gamma\left( {{y;{{2\;\mu} + A}},\sigma_{0}^{2}} \right)}} + {\frac{1}{4}{{\gamma\left( {{y;{{{- 2}\;\mu} + A}},\sigma_{0}^{2}} \right)}.}}}} & (59)\end{matrix}$

Hence, when μ is large enough,

$\begin{matrix}{P_{D,{opt}}^{y_{s}} = {{{\frac{1}{2}{Q\left( {- \frac{A}{\sigma_{0}}} \right)}} + \frac{1}{4}} = {0.6707.}}} & \;\end{matrix}$

Note that, as σ₀ decreases P_(D,opt) ^(y) ^(s) increases, i.e., betterdetection performance can be achieved by adding the optimal symmetricnoise.

Similarly, for the uniform noise case,

$\begin{matrix}{{C_{u}(x)} = {{\int_{- x}^{\infty}{{p_{u}(t)}\ {\mathbb{d}t}}} = \left\{ \begin{matrix}{0,} & {x < \frac{- a}{2}} \\{{\frac{x}{a} + \frac{1}{2}},} & {{- \frac{a}{2}} \leq x \leq 0.}\end{matrix} \right.}} & (60)\end{matrix}$

Substituting equation (60) for C_(s)(x) in (56) and taking thederivative w.r.t a, we have

$\begin{matrix}{\frac{\mathbb{d}P_{D}^{y_{s}}}{\mathbb{d}\alpha} = {{- \frac{1}{a^{2}}}{\int_{- \frac{a}{2}}^{0}{{{xH}(x)}\ {{\mathbb{d}x}.}}}}} & (61)\end{matrix}$

Setting it equal to zero and solving, we have a_(opt)=8.4143 in the pdfof uniform noise defined earlier. Additionally, we have P_(D,opt) ^(y)^(u) =0.6011.

For the Gaussian case, the optimal WGN level is readily determined since

$\begin{matrix}{P_{D}^{y_{s}} = {{\frac{1}{2}{Q\left( \frac{{- A} - \mu}{\sqrt{\sigma_{0}^{2} + \sigma^{2}}} \right)}} + {\frac{1}{2}{{Q\left( \frac{{- A} + \mu}{\sqrt{\sigma_{0}^{2} + \sigma^{2}}} \right)}.}}}} & (62)\end{matrix}$

Let σ² ₂=σ² ₀+σ² and take the derivative w.r.t σ₂ ² in equation (62),setting it equal to zero and solving, forming

$\begin{matrix}{{\sigma_{2}^{2} = {{2\frac{\mu\; A}{\ln\left( \frac{\mu + A}{\mu - A} \right)}} = 8.6562}},} & (63)\end{matrix}$and σ² _(opt)=σ² ₂−σ² ₀=7.6562, and correspondingly, P_(D,opt) ^(y) ^(g)=0.5807. Therefore, when σ² ₀<σ² ₂, adding WGN with variance σ² _(opt)can improve the detection performance to a constant level P^(yg)_(D,opt).

Table 1 below is a comparison of detection performance for different SRnoise enhanced detectors, and shows the values of P_(D,opt) ^(y) for thedifferent types of SR noise. Compared to the original data process withP_(D) ^(X)=0.5114, the improvement of different detectors are 0.1811,0.1593, 0.0897 and 0.0693 for optimum SR noise, optimum symmetric noise,optimum uniform noise and optimum Gaussian noise enhanced detectors,respectively.

SR Noise P_(n) ^(opt) P_(s) ^(opt) P_(u) ^(opt) P_(g) ^(opt) No SR NoiseP_(D) ^(y) .6915 .6707 .6011 .5807 .5115

FIG. 6 shows P_(D) ^(x) as well as the maximum achievable P_(D) ^(y)with different values of A. The detection performance is significantlyimproved by adding optimal SR noise. When A≦μ, a certain degree ofimprovement is also observed by adding suboptimal SR noise. When A issmall, x₀≈−μ and x₁≈μ, the detection performance of the optimum SR noiseenhanced detector is close to the optimum symmetric noise enhanced one.However, when A>0.6, the difference is significant. When A>μ=3, H(x)<0;∀x<0, so that P_(D,opt) ^(y) ^(s) =P_(D,opt) ^(y) ^(u) =P_(D,opt) ^(y)^(g) =P_(D) ^(x), i.e, the is optimal symmetric noise is zero (no SRnoise). However, by adding optimal SR noise, P_(D,opt) ^(y) is stilllarger than P_(D) ^(x), i.e., the detection performance can still beimproved. When A≧5, the P_(D) improvement is not that significantbecause P_(D) ^(x)>0.97≈1 which is already a very good detector.

The maximum achievable detection performance of different SR noiseenhanced detectors with different background noise σ₀ is shown in FIG.7. When σ₀ is small, for the optimum SR noise enhanced detectorsP_(D,opt) ^(y)≈1, while for the symmetric SR noise case P_(D,opt) ^(y)^(s) ≈0.75. When σ₀ increases, P_(D) ^(x) increases and the detectionperformance of SR noise enhanced detectors degrades. When σ₀≧σ₁, p₀(x)becomes a unimodal noise and the decision function φ is the same as thedecision function decided by the optimum LRT test given the false alarmP_(FA)=0.5. Therefore, adding any SR noise will not improve P_(D).Hence, all the detection results converge to P_(D) ^(x).

FIG. 8 compares the detection performance of different detectors w.r.t.μ when A=1 and σ₀=1 is fixed. P_(D) ^(x), P_(D,opt) ^(y) ^(u) andP_(D,opt) ^(y) ^(g) monotonically decrease when μ increases. Also, thereexists a unique μ value μ₀, such that when μ<μ₀ is small, p₀ is still aunimodal pdf, so that the decision function φ is the optimum one forP_(FA)=0.5. An interesting observation from FIG. 8 is that the P_(D) ofthe “optimum LRT” after the lowest value is reached, increases when μincreases. The explanation of this phenomenon is that when μ issufficiently large, the separation of the two peaks of the Gaussianmixtures increases as μ increases so that the detectability isincreased. When μ→∞, the two peaks are sufficiently separated, so thatthe detection performance of “LRT” is equal to the P_(D) when μ=0.

Finally, FIG. 9 shows the ROC curves for the detection problem when N=30and the different types of i.i.d SR noise determined previously areadded. Different degrees of improvement are observed for different SRnoise pdfs. The optimum SR detector and the optimum symmetric SRdetector performance levels are superior to those of the uniform andGaussian SR detectors and more closely approximate the LRT curve. ForLRT, the performance is nearly perfect (P_(D)≈1 for all P_(FA)s).

The present invention thus establishes a mathematical theory for thestochastic resonance (SR) noise modified detection problem, as well asseveral fundamental theorems on SR in detection theory. The detectionperformance of a SR noise enhanced detector is analyzed where, for anyadditive noise, the detection performance in terms of P_(D) and P_(FA)can be obtained by applying the expressions of the present invention.Based on these, the present invention established the conditions ofpotential improvement of P_(D) via the SR effect, which leads to thesufficient condition for the improvability/non-improvability of mostsuboptimal detectors.

The present invention also established the exact form of the optimal SRnoise pdf. The optimal SR noise is shown to be a proper randomization ofno more than two discrete signals. Also, the upper limit of the SRenhanced detection performance is obtained by the present invention.Given the distributions p₁ and p₀, the present invention provides anapproach to determine the optimal SR consisting of the two discretesignals and their corresponding weights. It should be pointed out thatthe present invention is applicable to a variety of SR detectors, e.g.,bistable systems.

The SR detectors that may be implemented with the present invention areshown in FIG. 10. For example, the nonlinear system block of FIG. 10 candepict the bistable system. Let x=[x₁, x₂, . . . , x_(n)]^(T) be theinput to the nonlinear system, and x′=[x₀₁, x₀₂, . . . , x_(N)]^(T) bethe output of the system as shown, where x′=f(x) is the appropriatenonlinear function. The decision problem based on x′ can be described bydecision function φ₀(•) as shown. It is easy to observe that thecorresponding decision function φ(•) for the ‘super’ detector (i.e., thenonlinear system plus detector) is φ(x)=φ₀(f(x)). Thus, the SR detectorscan be viewed as the system in FIG. 10 without the additive SR noise n.To summarize, the present invention admits conventional SR systems andallows improved detection system by adding n as shown in FIG. 10.

FIG. 11 illustrates a diagram of a SR detection system obtained by amodification of the observed data, x. The statistical properties of thedata are changed by adding independent SR noise n to yield a new processy such that y=x+n. This process, in turn is provided as input to thenoise modified detector.

Based on the mathematical framework of the present invention, for aparticular detection problem, the detection performance of six differentdetectors are compared, namely, the optimum LRT detector, optimum noiseenhanced sign detector, optimum symmetric noise enhanced sign detector,optimum uniform noise enhanced sign detector, optimum Gaussian noiseenhanced sign detector and the original sign detector. Compared to thetraditional SR approach where the noise type is predetermined, muchbetter detection performance is obtained by adding the proposed optimumSR noise to the observed data process. The present invention thuscorresponds with the observed SR phenomenon in signal detectionproblems, and greatly advances the determination the applicability of SRin signal detection. The present invention can also be applied to manyother signal processing problems such as distributed detection andfusion as well as pattern recognition applications.

The present invention may thus be used to increase the probability ofdetecting signals embedded in non-Gaussian noise. The first step is torecord data from an observed data process. Next, stochastic resonancenoise is added to said recorded. The appropriate stochastic resonancenoise is controlled by determining the stochastic resonance noiseprobability density function (PDF) that does not increase the detectorprobability of false alarm.

The SR noise may be determined for the case of a known data probabilitydensity function by determining from the known probability densityfunction of the observed data process the stochastic resonance noise PDFthat equals λδ(n−n₁)+(1−λ)δ(n−n₂), with values n₁ and n₂ equal to thoseof the two delta function locations, and with probabilities equal to λand (1−λ), respectively. More specifically, the stochastic resonancenoise PDF may be calculated by determining F_(i)(x)=∫_(R) _(N)φ(y)p_(i)(y−x)dy i=0, 1, using known critical function φ(y) and knowndata probability density functions p_(i)(•), i=0, 1; determining thethree unknown quantities n₁, n₂, and λ using the known values k₀, f₀₁and f₀₂ and the following three equations:

$\begin{matrix}{{{\frac{\mathbb{d}J}{\mathbb{d}f_{0}}\left( {f_{01}\left( k_{0} \right)} \right)} = {\frac{\mathbb{d}J}{\mathbb{d}f_{0}}\left( {f_{02}\left( k_{0} \right)} \right)}};} & (i) \\{{{\frac{\mathbb{d}J}{\mathbb{d}f_{0}}\left( {f_{02}\left( k_{0} \right)} \right)} = k_{0}};} & ({ii})\end{matrix}$(iii) J(f₀₂(k₀))−J(f₀₁(k₀)=k₀(f₀₂(k₀)−f₀₁(k₀)); and determining theprobability of occurrence for n₁ and n₂ as λ and 1−λ, respectively,using the equation

$\lambda = {\frac{{f_{02}\left( k_{0} \right)} - P_{FA}^{x}}{{f_{02}\left( k_{0} \right)} - {f_{01}\left( k_{0} \right)}}.}$

Alternatively, the SR noise for the case of a known data probabilitydensity function may be calculated by determining the stochasticresonance noise PDF that consists of a single delta function, δ(n−n₀)with value n₀ equal to the delta function location with probability one.The minimum probability of error may be calculated from

$P_{e,\min} = {\pi_{1}\left\lbrack {1 - {\max\limits_{f_{0}}{G\left( {f_{0},\frac{\pi_{0}}{\pi_{1}}} \right)}}} \right\rbrack}$where G(f₀, k)=J(f₀)−kf₀=kf₀=P_(D)−kP_(FA). The single delta functionlocated at n₀ is calculated from n₀=F₀ ⁻¹(f₀), where f₀ is the value themaximizes

${G\left( {f_{0},\frac{\pi_{0}}{\pi_{1}}} \right)}.$

The SR noise for the case of labeled data with an unknown data PDF maybe determined by first calculating the stochastic resonance noise PDFthat consists of two delta functions. This step is accomplished byestimating the stochastic resonance noise consisting of two randomvariables n₁ and n₂ by using many algorithms, such asexpectation-maximization (EM) and the Karzen method to estimate theunknown data PDFs, and applying the estimated PDFs and the stochasticresonance noise PDF may be calculated by determining F_(i)(x)=∫_(R) _(N)φ(y)p_(i)(y−x)dy i=0, 1, using known critical function φ(y) and knowndata probability density functions p_(i)(•), i=0, 1; determining thethree unknown quantities n₁, n₂, and λ using the known value k₀, andestimated values {circumflex over (f)}₀₁, {circumflex over (f)}₀₂, and Ĵin the following three equations:

$\begin{matrix}{{\frac{\mathbb{d}\hat{J}}{\mathbb{d}f_{0}}\left( {{\hat{f}}_{01}\left( k_{0} \right)} \right)} = {\frac{\mathbb{d}\hat{J}}{\mathbb{d}f_{0}}\left( {{{\hat{f}}_{02}\left( k_{0} \right)};} \right.}} & (i) \\{{{\frac{\mathbb{d}\hat{J}}{\mathbb{d}f_{0}}\left( {{\hat{f}}_{02}\left( k_{0} \right)} \right)} = k_{0}};} & ({ii})\end{matrix}$(iii) Ĵ({circumflex over (f)}₀₂(k₀))−Ĵ({circumflex over(f)}₀₁(k₀)=k₀({circumflex over (f)}₀₂(k₀)−{circumflex over (f)}₀₁(k₀));and determining the probability of occurrence for n₁ and n₂ as λ and1−λ, respectively, using the equation

$\lambda = {\frac{{{\hat{f}}_{02}\left( k_{0} \right)} - P_{FA}^{x}}{{{\hat{f}}_{02}\left( k_{0} \right)} - {{\hat{f}}_{01}\left( k_{0} \right)}}.}$

The next step is to determine the stochastic resonance noise consistingof two random variables n₁ and n₂ with values equal to those of the twodelta function locations and with probabilities equal to those of saidstochastic resonance noise probability density function; adding saidstochastic resonance noise to said data; applying said fixed detector tothe resulting data process.

Finally, a test statistic for signal detection is calculated under aconstant probability of false alarm rate (CFAR) constraint, such thatthe performance of suboptimal, nonlinear, fixed detectors operating insaid non-Gaussian noise are improved. Increasing the probability ofdetecting signals embedded in non-Gaussian noise comprises adding thestochastic resonance noise n₁ and n₂ with probability λ and 1−λ,respectively, to the data, and applying the fixed detector to theresulting data process.

The present invention also provides a method for evaluating functionsusing f₁, J(f₀), and

$\frac{\mathbb{d}J}{\mathbb{d}f_{0}}$where for any f₀, the equation x=F₀ ⁻¹(f₀) is solved, and the value off₁ is obtained by

${f_{1} = {F_{1}(x)}},{{J\left( f_{0} \right)} = {\max\limits_{f_{1}}\left( {f_{1}\left( f_{0} \right)} \right)}},{and}$$\frac{\mathbb{d}J}{\mathbb{d}f_{0}} = {\lim\limits_{\Delta\rightarrow 0}{\frac{{J\left( {f_{0} + \Delta} \right)} - {J\left( f_{0} \right)}}{\Delta}.}}$

Following is background information relative to the formulas of thepresent invention and the applicable theorems on convex functions andconvex sets.

A. Convex Functions

A function f: I→R is called convex iff(λx+(1−λ)y)≦λf(x)+(1−λ)f(y)  (64)for all x, y∈I and λ in the open interval (0,1). It is called strictlyconvex provided that the inequality (64) is strict for x≠y. Similarly,if −f: I→R is convex, then we say that f: I→R is concave.

Theorem A-1: Suppose f″ exists on (a,b). Then f is convex if and only iff″(x)≧0. And if f″(x)>0 on (a,b), then f is strictly convex on theinterval.

B. Convex Sets

Let U be a subset of a linear space L. We say that U is convex if x, y∈Uimplies that z=[λx+(1−λ)y]∈U for all λ∈[0,1].

Theorem A-2: A set U⊂L is convex if and only if every convex combinationof points of U lies in U.

We call the intersection of all convex sets containing a given set U theconvex hull of U denoted by H(U).

Theorem A-3: For any U⊂L, the convex hull of U consists precisely of allconvex combinations of elements of U.

Furthermore, for the convex hull, we have Carathéodory's theorem forconvex sets.

Theorem A-4 (Carathéodory's Theorem): If U⊂L, and its convex hull ofH(U) has dimension m, then for each z∈H(U), there exists m+1 points x₀,x₁, x_(m) of U such that z is a convex combination of these points.

As discussed herein, a systematic approach to enhance the performance ofsuboptimal signal detection and estimation systems by adding suitablenoise to the input signal has been developed (suboptimality may be dueto inaccurate statistical models, model mismatch and system limitation,such as fixed decision threshold). This type of phenomenon is also knownas stochastic resonance (SR) which occurs in some non-linear systemswhere the signals can be enhanced by adding suitable noise under certainconditions. Under the Bayesian and Neyman-Pearson frameworks, it hasbeen determined whether or not a given detector is improvable by theaddition of noise. If the detector is improvable, the PDF of the optimumnoise to be added has been determined. See H. Chen, P. K. Varshney, J.H. Michels, and S. M. Kay, “Theory of the stochastic resonance effect insignal detection: Part 1—fixed detectors,” IEEE Trans. on SignalProcessing, vol. 55, no. 7, pp. 3172-3184, July, 2007 A number of casesincluding fixed and variable detectors were considered and significantperformance enhancement was shown. H. Chen et al. supra; Chen et al. andH. Chen and P. K. Varshney, “Theory of the stochastic resonance effectin signal detection: Part II—variable detectors,” IEEE Trans. on SignalProcessing, October, 2008.

In accordance with an embodiment of the present invention, a morespecific matter where SR noise is employed to enhance the detection ofmicro-calcifications in mammograms is also presented herein. A SRnoise-based detection algorithm and a general detection enhancementframework to improve the performance of the suboptimal detectors hasbeen developed. The dependence of the determination of the optimum SRnoise is reduced on the knowledge of the PDFs of the object (lesion) andbackground (normal tissues) by employing iterative learning procedures.An iterative SR noise-based detection enhancement scheme with memory toimprove the efficiency and robustness of the SR noise-based detectionsystems is also developed. Moreover, a more general SR noise-baseddetection enhancement framework is presented. Experiments and analysesare carried out to compare the performance of the presented SRnoise-based detection algorithms and several other detection andclassification methods. Detection algorithms considered/developed in thefollowing experiments include the Gaussian background assumption-baseddetector, General Gaussian detector, General Gaussian detector-basediterative detector, Gaussian assumption-based dynamic clusteringalgorithm, iterative mode separation algorithm, Gaussian mixturemodel-based clustering method, and higher order statistics method basedon local maxima detection and adaptive wavelet transform. Thedescriptions of these algorithms are presented and discussed in thefollowing Examples presented below. The equations referenced in thefollowing Examples start with Equation No. 1.

Advantages of the invention are illustrated by the following Examples.However, the particular materials and amounts thereof recited in theseexamples, as well as other conditions and details, are to be interpretedto apply broadly in the art and should not be construed to undulyrestrict or limit the invention in any way.

EXPERIMENTAL DATA

The mammograms used herein are from the Digital Database for ScreeningMammography (DDSM), and the Mammographic Image Analysis Society (MIAS)Mini-mammographic Database. However, the majority of the mammograms usedin the experiments are from DDSM, and only few of them are from the MIASdatabase, so the experimental parameters, such as the processing windowsize, are determined mainly based on DDSM.

DDSM has 2620 cases available in 43 volumes. A case consists of between6 and 10 mammograms, where the grey levels are quantized to 16 bits, andresolution of the images is 50 microns/pixel. The MIAS Database includes330 mammograms with the resolution being 200 microns/pixel. The locationand types (malignant or benign) of the mammogram lesions are identifiedby expert radiologists and used as the ground truth in our work. Asdiscussed herein, the emphasis is on location detection based on theground truth.

Three types of representative abnormal mammograms withmicro-calcifications (clusters) including one having homogeneousbackground with a small number of isolated micro-calcifications werechosen, one having homogeneous background with a large number ofmicro-calcifications (clusters) and one having inhomogeneous backgroundwith a moderate number of micro-calcifications (clusters), respectively.These three types of mammograms cover a broad spectrum of mammogrammicro-calcification (cluster) cases. Seventy five images selected fromthe three types of mammograms, 25 for each type, are employed to testthe algorithms.

Micro-calcifications are very small, their sizes are in the range of0.05-1.00 mm, and the average is 0.3 mm. Those smaller than 0.1 mmcannot be easily distinguished in the film-screen mammography from thehigh-frequency noise. The width of the majority of themicro-calcifications in these experiments is in the range between 0.25and 0.5 mm. A micro-calcification cluster is considered to be a group of3 to 5 or more micro-calcifications, 5 mm apart. The processing windowsize of 49 by 49 was chosen, which is based on experiments that wereconducted as well as the characteristics and the size of the lesions.The experiments also indicated that the detection results were not verysensitive to the choice of window size provided that the window size wasin the range between 31 and 61 when processing the data.

Since pixel by pixel detection was carried out, any isolated detectedpositive should not be considered to be a lesion due to themicro-calcification size mentioned above as well as the fact that thehigh-frequency noise may have serious influence on an individual pixel.Therefore, a micro-calcification (cluster) is declared to be detectedonly if at least 4 by 4 positively detected pixels are in a clump.

Example 1 Problem Statement And Gaussian Assumption-Based LesionDetection

This Example develops three lesion detectors based on Gaussianassumption. As discussed in more detail below, it will be shown that thedetectors are all suboptimal detectors, suffering from model mismatch.

In this Example, the lesion detection problem from a statisticalhypothesis testing point of view is introduced, and then three Gaussianassumption-based detectors for the lesion detection task are presented.It will be shown via experiments that the performance of these detectorsis not satisfactory due to the large number of false alarms. This is dueto model mismatch, and it results in suboptimality of the detectors.

Higher pixel intensity than the surrounding normal tissues distinguisheslesions from the normal structures in mammograms, which is one of themost important features of abnormal mammograms. The novel algorithmsdeveloped and presented herein perform the detection by exploring thepixel intensity information. This anomaly detection problem is delt withby using statistical hypothesis testing methods. Formally, a choiceshould be made between one of the two hypotheses corresponding to theabsence and presence of micro-calcifications on a pixel-by-pixel basis,H ₀ : y[m]=ω[m]H ₁ : y[m]=s[m]+ω[m]  (1)where m is the pixel index corresponding to the pixel observation underconsideration, y[m] is the observed pixel intensity, larger than orequal to zero, s[m] is the lesion signal, and w[m] is the backgroundnoise that is assumed to obey Gaussian distribution with mean μ_(b) andvariance σ_(b) ². The noise is assumed to be additive, independentlydistributed and independent of the noise-free mammogram data. Aprocessing window is employed with the pixel under consideration beingat the center of the window. This window is employed to estimate theparameters of the detector by using the pixels included in it.A. Gaussian Background Assumption-Based Detector (GBAD)

The Gaussian background assumption leads to linear and tractablesolutions. The micro-calcifications, the signals of interest here, arebrighter spots than the surrounding normal background tissues. So themicro-calcification is modeled as a signal with constant amplitude, andthe lesion detection problem is to detect a DC signal in Gaussian noise,which we refer to as the Gaussian background assumption-based detector(GBAD). The lesion signal s[m] in Eq. (1) is, therefore, a constantintensity.

For this anomaly detection problem, the a priori probabilities of thebackground and lesion pixels are unavailable, so the Neyman-Pearsoncriterion for the detection task is employed. It is well known thatunder the Neyman-Pearson criterion, the optimal detector is a likelihoodratio test given as

$\begin{matrix}{{L\left( \overset{\_}{y} \right)} = {\frac{p\left( {\overset{\_}{y};H_{1}} \right)}{p\left( {\overset{\_}{y};H_{0}} \right)}\overset{{decide}\mspace{14mu} H_{1}}{>}\gamma}} & (2)\end{matrix}$where y is the observation vector, p( y;H₁) and p( y;H₀) are the PDFsunder hypotheses H₁ and H₀, respectively. The threshold γ is found fromP _(F)=∫_({ y:L( y)>γ) p( y;H ₀)d y=β  (3)where β is the desired value of the P_(F).

Under the Gaussian background and DC signal assumptions, p( y;H₁) and p(y; H₀) all obey Gaussian distribution with the same variance σ_(b) ²,and the optimal test given in Eq. (2) can be expressed in terms of theGBAD test statistic T_(GBAD)(y) as follows

$\begin{matrix}{{T_{GBAD}(y)} = {{y - \mu_{b}}\overset{{decide}\mspace{14mu} H_{1}}{>}\gamma_{1}}} & (4)\end{matrix}$where y is the intensity of the pixel under consideration, and thethreshold γ₁ is determined from the desired P_(F) and the statisticalparameters, i.e., mean and variance, of the pixels in the processingwindow.

To estimate the detector's parameters, an initial detection is firstcarried out in the processing window to perform a coarse detection, andthe resulting detected negatives (H₀) and positives (H₁) are employed toestimate the parameters. Many methods, such as a local maxima filter oradaptive thresholding techniques, can perform the initial detection.Local maxima filter is employed in this work because the lesion pixelsgenerally have a higher intensity than the surrounding normal backgroundtissues.

For cancer diagnosis, the most serious mistake is to miss any lesions.To reduce the probability of miss (P_(M)), a “safer” initial detectionis used and an attempt is made to exclude all the lesion pixels from thebackground. It can be realized by using a local maxima filter withappropriate window size and local threshold, permitting more pixelshaving relatively higher intensities in the local regions to beclassified into the lesion part.

The detection results for GBAD are shown in FIGS. 13 (b), 14 (b) and 15(b), corresponding to three types of mammograms withmicro-calcifications, as discussed further below. From the figures, itcan be seen that the micro-calcifications are completely detected butwith a higher P_(F) than the desired value, 0.01 used in theexperiments. At this point, it suffices to say that the performance ofthe detector is not satisfactory. A detailed discussion of theexperimental results is postponed to Example 5, infra.

B. General Gaussian Detector (GGD)

Micro-calcifications, especially micro-calcification clusters, have asmall size but generally do not have a constant intensity, so a Gaussianmodel as opposed to a constant signal model is proposed in this sectionto be a more reasonable model to represent the signal part. Thus, theproblem can be considered to be the problem of detection of Gaussiansignals in Gaussian noise. This detector is referred to as the generalGaussian detector (GGD). The lesion signal s[m] in Eq. (1) under thisassumption obeys Gaussian distribution, i.e., s[m]˜N(μ_(s),σ_(s) ²).

The detected positive pixels (corresponding to lesions) and negativepixels (corresponding to the background) in the initial detection areemployed to coarsely estimate the means and variances of the lesion andbackground pixel intensity PDFs. Under the GGD assumption, p( y; H₁) andp( y; H₀) in Eq. (2) obey Gaussian distribution but with differentvariances, and the optimal test is still the likelihood ratio test givenin Eq. (2). The optimum test can be expressed in terms of the GGD teststatistic T_(GGD)(y) as follows

$\begin{matrix}{{T_{GGD}(y)} = {{{\frac{\sigma_{s}^{2}}{\sigma_{b}^{2}}\left( {y - \mu_{b}} \right)^{2}} + {2{\mu_{s}\left( {y - \mu_{b}} \right)}}}\overset{{decide}\mspace{14mu} H_{1}}{>}\gamma_{2}}} & (5)\end{matrix}$where the threshold γ₂ is determined from the desired P_(F). Thestatistical parameters, namely the means and the variances, of theinitially detected positive and negative pixels can be estimated usingthe processing window with the pixel under consideration at the centerof the window.

It can be seen that when σ_(s) ²→0, the first term on the right side ofEq. (5) tends to zero, and Eq. (5) reduces to a form similar to Eq. (4),which corresponds to s[m] being a constant signal. It is also noticedthat Eq. (5) is a detector with two thresholds because the teststatistic is quadratic. Due to the nature of the abnormal mammograms,i.e., lesion pixels have intensities that are generally higher than thesurrounding normal background tissues and the probability of theintensities of lesion pixels falling below the lower threshold isextremely small, thus, only the higher threshold is employed to classifythe mammogram pixels into background and lesions. Therefore, the higherthreshold of the test in Eq. (5) is used herein for the detection task.

The detection results for GGD are shown in FIGS. 13 (c), 14 (c) and 15(c), as discussed further in Example 5, infra, where all themicro-calcifications are discovered by GGD, with less false positivescompared with the GBAD.

C. GGD-Based Iterative Detector (GGD_ID)

Encouraged by the improvement achieved by the GGD over GBAD, aniterative method is proposed to further improve the performance of GGDby increasingly improving the estimation of statistical parameters in aniterative manner.

At each step of the iteration, the GGD is designed with the parameters,μ_(s), μ_(b), σ_(s) ² and σ_(b) ², corresponding to the background andmicro-calcifications, estimated from the detection result in thepreceding iteration as opposed to keeping them fixed during alliterations, which results in different thresholds at each iteration.

The procedure of the iterative detection algorithm is described asfollows:

Initialization: Initial detection using the coarse detector described inExample 1. A.

-   Step 1: Means and variances of the detected positive (lesion) and    negative (background) pixels are calculated.-   Step 2: Detection is performed using the GGD (Eq. (5)) with the    desired P_(F) and the updated parameters, μ_(s), μ_(b), σ_(s) ² and    σ_(b) ², calculated in Step 1. If there are no differences in the    detected positives and negatives between two successive detections,    terminate the algorithm, else go to Step 1.

The presented GGD_ID algorithm is similar in spirit to Gaussianmodel-based dynamic clustering (GMDC), in which both background andlesions are assumed to obey Gaussian distributions, and the detection(or clustering) and parameter updating are performed in an iterativemanner. The difference is that the method presented here incorporates anadditional constraint in terms of the desired value of P_(F). The reasonP_(F) is included in the algorithm is that at each step of theiteration, some detected negative pixels have intensities much largerthan the mean of the detected background pixels and are close to that ofthe detected lesion pixels. In other words, some pixels have anon-negligible and, in fact, fairly high probability to belong to thelesion part. Since it is preferable that no lesions be missed, thesepixels are classified into the lesion part by the desired P_(F) value,such as the value 0.01 used herein. It is discussed in Example 5, infra,that the iterative detection method presented here performs better thanthe GMDC.

The detection results for this detector are shown in FIGS. 13 (d), 14(d) and 15 (d), and discussed in Example 5, infra, where the GGD isemployed iteratively four times on the mammograms. Experiments show thatthe method generally converges within 5 iterations.

D. Model Mismatch Analysis

From the experimental results, it can be seen that the detectionperformance has improved with the melioration of the detection schemes,but the final results are still not satisfactory as seen via inexactlesion contours and large number of false positives. The resultingdiagnosis may result in additional testing and biopsies for spots onmammograms that finally turn out to be harmless, which is a weaknessmany CAD systems exhibit currently.

One major reason for the unsatisfactory detection is that the Gaussianassumption does not accurately model the background distribution and theresulting test including the detection threshold is not optimal. A moreaccurate model for the background, heavy-tailed symmetric α stable (SαS)distribution, was proposed in Banerje et al., which has also beenverified by the experiments described herein. Amit Banerje and RamaChellappa, “Tumor detection in digital mammograms,” in Proc.International Conference on Image Processing (ICIP'00), vol. 3, pp.432-435, Vancouver, BC, Canada, 2000.

For verification, the following is shown in FIG. 12—the amplitudeprobability distribution (APD) plots of real-world mammogram backgrounddata of a mammogram from the MIAS Mini-mammographic Database, andsimulated Gaussian distribution and heavy-tailed SαS distribution dataon a log-log scale (showing that the mammogram pixel intensities obeyheavy-tailed SαS distribution more closely). Plotting APD is a commonlyused method to test impulsive noise. It is defined as the probabilitythat the noise amplitude is above some threshold. It can be seen fromFIG. 12 that for small amplitudes, the simulated heavy-taileddistribution and Gaussian distribution provide good fits to themammogram data. At larger amplitudes (i.e., at the tails), the simulatedheavy-tailed SαS distribution is shown to be a better fit than theGaussian one. In addition, the plots of the mammogram data and thesimulated heavy-tailed SαS data decay linearly with a constant slopecompared with that of the Gaussian data. These two observations indicatethat the heavy-tailed SαS distribution is a better model than theGaussian model for the background pixel intensities of a digitalmammogram. Hence, there exists empirical support for the existence ofthe SαS noise distribution in mammogram background (as opposed to theGaussian distribution). Theoretical analysis and more detaileddiscussion on this can be found in Banerje et al., supra.

One approach to the design of the optimal lesion detector is to derivethe optimal test under the Neyman-Pearson formulation when thebackground is modeled as the SαS distribution. However, the difficultiesin learning the parameters of the SαS distribution from the real-worlddata as well as the off-line integration when calculating the detectionthreshold constrains the practical application of the optimal SαS-baseddetectors. In the following Examples, an alternate approach inaccordance with an embodiment of the present invention is investigated,namely the application of SR noise, to the lesion detection problem. Thesuboptimal detectors designed based on the Gaussian noise backgroundassumption will continue to be used. Admittedly there is a modelmismatch. In an attempt to overcome the deterioration in the detectorperformance, SR noise at the input will be added to the detector. Itwill be seen that the SR noise-based detector yields significantperformance enhancement and is easy to implement.

Example 2 Optimum SR Noise-Enhanced Signal Detection

This Example presents the main results of work on SR noise-enhancedsignal detection under Neyman-Pearson criterion, where the optimum formof the SR noise is determined.

One of the main goals of an embodiment of the present invention is todevelop SR noise-enhanced detection methods for lesion detection inmammograms. First, the fundamental results on SR noise-enhanced signaldetection is presented in this Example.

A binary statistical decision problem is considered. Similar to thestatement in Example 1, a choice between the two hypotheses should bemade

$\begin{matrix}\left\{ \begin{matrix}{{H_{0}:{p_{\overset{\_}{y}}\left( {\overset{\_}{y};H_{0}} \right)}} = {p_{0}\left( \overset{\_}{y} \right)}} \\{{H_{1}:{p_{\overset{\_}{y}}\left( {\overset{\_}{y};H_{1}} \right)}} = {p_{1}\left( \overset{\_}{y} \right)}}\end{matrix} \right. & (6)\end{matrix}$where y is an N-dimensional data vector, i.e., y∈R^(N). p₀( y) and p₁(y) are the pdfs of y under H₀ (background) and H₁ (lesion) hypotheses,respectively. Pixel by pixel detection is only considered herein, so yreduces to a scalar y. During the decision process, a test is necessaryto choose between the two hypotheses, which can be completelycharacterized by a critical function, or decision function, φ( y), 0≦φ(y)≦1, and

$\begin{matrix}{{\phi\left( \overset{\_}{y} \right)} = \left\{ \begin{matrix}{1:{{T\left( \overset{\_}{y} \right)} > \gamma}} \\{{\beta:{T\left( \overset{\_}{y} \right)}} = \gamma} \\{0:{{T\left( \overset{\_}{y} \right)} < \gamma}}\end{matrix} \right.} & (7)\end{matrix}$where T is the test statistic and a function of y. γ is the threshold,and 0≦β≦1 is a suitable number.

The detection performance of this test can be evaluated in terms ofP_(D) and P_(F),

$\begin{matrix}{P_{D}^{\overset{\_}{y}} = {\int_{R^{N}}{{\phi\left( \overset{\_}{y} \right)}{p_{1}\left( \overset{\_}{y} \right)}{\mathbb{d}\overset{\_}{y}}}}} & (8) \\{P_{F}^{\overset{\_}{y}} = {\int_{R^{N}}{{\phi\left( \overset{\_}{y} \right)}{p_{0}\left( \overset{\_}{y} \right)}{\mathbb{d}\overset{\_}{y}}}}} & (9)\end{matrix}$where P_(D) ^(y) and P_(F) ^(y) represent the P_(D) and P_(F) of thedetector based on the input y, respectively.

The SR noise-based detection enhancement scheme is to add an appropriatenoise n to the original data y, which yields a new data vector zz= y+ n   (10)where n is either a random vector with pdf p _(n) (.) or a nonrandomsignal.

The binary hypotheses testing problem for this new observed data can beexpressed as

$\begin{matrix}\left\{ \begin{matrix}{{H_{0}:{p_{\overset{\_}{z}}\left( {\overset{\_}{z};H_{0}} \right)}} = {{p_{0}\left( \overset{\_}{z} \right)} = {\int_{R^{N}}{{p_{0}\left( \overset{\_}{y} \right)}{p_{\overset{\_}{n}}\left( {\overset{\_}{z} - \overset{\_}{y}} \right)}{\mathbb{d}\overset{\_}{y}}}}}} \\{{H_{1}:{p_{\overset{\_}{z}}\left( {\overset{\_}{z};H_{1}} \right)}} = {{p_{1}\left( \overset{\_}{z} \right)} = {\int_{R^{N}}{{p_{1}\left( \overset{\_}{y} \right)}{p_{\overset{\_}{n}}\left( {\overset{\_}{z} - \overset{\_}{y}} \right)}{\mathbb{d}\overset{\_}{y}}}}}}\end{matrix} \right. & (11)\end{matrix}$

The SR noise-enhanced fixed detectors are considered herein whoseparameters, such as the thresholds, are unchanged before and afteradding the SR noise, so the critical function φ of z is the same as thatof y. Therefore,

$\begin{matrix}\begin{matrix}{P_{D}^{\overset{\_}{z}} = {\int_{R^{N}}{{\phi\left( \overset{\_}{z} \right)}{p_{1}\left( \overset{\_}{z} \right)}{\mathbb{d}\overset{\_}{z}}}}} \\{= {\int_{R^{N}}{{p_{\overset{\_}{n}}\left( \overset{\_}{y} \right)}\left( {\int_{R^{N}}{{\phi\left( \overset{\_}{z} \right)}{p_{1}\left( {\overset{\_}{z} - \overset{\_}{y}} \right)}{\mathbb{d}\overset{\_}{z}}}} \right){\mathbb{d}\overset{\_}{y}}}}} \\{= {\int_{R^{N}}{{F_{1}\left( \overset{\_}{y} \right)}{p_{\overset{\_}{n}}\left( \overset{\_}{y} \right)}d\overset{\_}{y}}}}\end{matrix} & (12)\end{matrix}$

And similarly

$\begin{matrix}{P_{F}^{\overset{\_}{z}} = {{\int_{R^{N}}{{\phi\left( \overset{\_}{z} \right)}{p_{0}\left( \overset{\_}{z} \right)}{\mathbb{d}\overset{\_}{z}}}} = {\int_{R^{N}}{{F_{0}\left( \overset{\_}{y} \right)}{p_{\overset{\_}{n}}\left( \overset{\_}{y} \right)}{\mathbb{d}\overset{\_}{y}}}}}} & (13)\end{matrix}$where

${F_{i}\left( \overset{\_}{y} \right)} = {\int_{R^{N}}{{\phi\left( \overset{\_}{z} \right)}{p_{i}\left( {\overset{\_}{z} - \overset{\_}{y}} \right)}{{\mathbb{d}\overset{\_}{y}}.}}}$corresponds to hypothesis H_(i).

The sufficient condition for improvability of detection via SR noise isgiven in Theorem 1.

-   -   Theorem 1: If J(P_(F) ^(y) )>P_(D) ^(y) or J″(P_(F) ^(y) )>0        when J(t) is second-order continuously differentiable around        P_(F) ^(y) , then there exists at least one noise process n with        PDF p _(n) (.) that can improve the detection performance, where        J(t) is defined as the maximum value of f₁ given f₀, i.e.,        J(t)=sup(f₁: f₀=t). f₀ and f₁ are the given values of F₀ and F₁,        respectively.

Theorem 2 determines the form of the optimum SR noise when the detectoris improvable.

-   -   Theorem 2: To maximize P_(D) ^(z) , under the constraint that        P_(F) ^(z) ≦P_(F) ^(y) , the optimum noise can be expressed as p        _(n) ^(opt)( n)=λδ( n− n ₁)+(1−λ)δ( n− n ₂), where λ and 1−λ are        the occurrence probabilities of the suitable N-dimensional        vectors n ₁ and n ₂, 0≦λ≦1.

The approach to determine λ, n ₁ and n ₂ is discussed in detail in H.Chen, P. K. Varshney, J. H. Michels, and S. M. Kay, “Theory of thestochastic resonance effect in signal detection: Part 1—fixeddetectors,” IEEE Trans. on Signal Processing, vol. 55, no. 7, pp.3172-3184, July, 2007. They can be determined in practice usingnumerical methods. Since the optimum SR noise is a randomization of twodeterministic vectors, it is called the “Two peak SR noise” herein.

The advantage of a SR noise-enhanced fixed detector is that theparameters, such as the threshold, of the original detector do not needto be changed, yet better detection performance is expected. In otherwords, model mismatch can be handled fairly easily by using thisapproach. However, to obtain the optimum SR noise, full knowledge of thepdfs under the two hypotheses is required, which in real-worldapplications is generally not available. In the next two Examples, theissue of how to find the suitable SR noise for enhancing a suboptimallesion detector when the knowledge of the PDFs is incomplete isdiscussed.

Example 3 SR Noise-Enhanced Gaussian Assumption-Based Detection

This Example relates to the development of a SR noise-based detectionalgorithm for lesion detection that attempts to improve the suboptimaldetectors discussed in Example 1. An interative detection schemeinvolving the use of SR noise with memory is also presented.

In this Example, the SR noise-enhanced detection approach is employedfor finding lesions and enhancing the previously discussed suboptimaldetectors based on the Gaussian assumption. Pixel-by-pixel detection wasperformed. The suboptimal detectors to be improved result from the modelmismatch and the lack of information about the mammogram statistics.These detectors are excellent candidates for the application of the SRnoise-enhanced detection scheme.

The basic idea of the SR noise-enhanced detection is to obtain theoptimum additive SR noise based on the knowledge of the PDFs of thelesion and the background signals. Since these PDFs are not known, theyneed to be estimated from the given mammogram itself. The mammogram ismodified with the optimum additive SR noise determined using theestimated pdf, and then the original suboptimal detector performs thedetection. Two SR noise-based schemes are presented in this Example forimproving lesion detection.

A. Two Peak SR Noise-Enhanced Gaussian Background Assumption-BasedDetection (2SR-GBAD)

In this algorithm, an attempt is made to reduce the dependence of the SRnoise determination on the knowledge of the true pdfs and increasinglyenhance the suboptimal detectors through an iterative procedure.

The SR noise is first used to enhance the GBAD discussed in Example 1.To achieve this goal, the coarse detection of the lesion and backgroundwas performed using the local maxima filter mentioned in Example 1A. Thedetection threshold is calculated for the GBAD, which is suboptimum dueto model mismatch. Then, the probability densities under H₁ and H₀ areobtained using the kernel density estimation method based on thedetected positives and negatives. Kinosuke Fukunaga, Introduction toStatistical Pattern Recognition (second edition), Academic Press,September, 1990. The parameters of the SR noise are calculated from thesuboptimum threshold and the estimated densities. The SR noise is addedto the original mammogram. Detection is performed on the SRnoise-modified data using the original detector. This procedure isrepeated in an iterative manner until the difference between twosuccessive detection results is very small (There could be many methodsto define and evaluate the difference. In these experiments, thedifference is defined as the ratio of the number of differently labeledpixels in two successive detections to the total number of pixels in themammogram. The labeled pixel here means a pixel classified as a positive(lesion) pixel or a negative (background) pixel. The iterative processis terminated when the ratio is smaller than a desired value.).

The procedure of the 2SR-GBAD detection algorithm is described asfollows:

Initialization: Initial detection using the coarse detector described inExample 1 A.

-   Step 1: Mean μ_(b) and variance σ_(b) ² of the background are    estimated, based on the detected negative pixels. The detection    threshold is updated based on the desired P_(F) as well as μ_(b) and    σ_(b) ² using Eq. (3), where it is assumed that the background obeys    Gaussian distribution (see GBAD in Example 1A).-   Step 2: The pixels are detected with the updated threshold found in    Step 1. The resulting detected positive and negative pixels are    employed for estimating probability densities under the two    hypotheses using the kernel density estimation method.-   Step 3: The updated threshold in Step 1 and the newly estimated    probability densities in Step 2 are used to determine the SR noise    with the method mentioned in Example 2.-   Step 4: The mammogram data is modified by adding to the original    pixel intensities the SR noise determined in Step 3.-   Step 5: Detection is performed with the detector updated in Step 1    using the modified data from Step 4. If the difference between two    successive detection results is very small, terminate the algorithm    else go to Step 1.

According to the experiments, a good initialization can be generated byschemes such as a maxima filter with an appropriate window size andthreshold, such that satisfactory detection can still be obtained evenwhen the threshold update procedure in Step 1 is not performed duringthe iterations. In other words, the critical function can remain fixedthroughout the iterations if the initial detection is good enough. Thethreshold updating can also be performed every several iterations toimprove the computation speed.

In a similar manner, the above procedure can be used to design the SRnoise enhanced GGD test, i.e., 2SR GGD where the means, μ_(s) and μ_(b),and variances, σ_(s) ² and σ_(b) ², of the detected positives andnegatives as well as the desired P_(F) are used together to update thethreshold in Step 1. The rest of the four steps of 2SR-GGD are the sameas those of 2SR-GBAD. Since GGD is a more accurate model for abnormalmammograms, which can be seen in the comparison between the detectionresults of GBAD and GGD, 2SR-GGD yields better performance than2SR-GBAD, according to the experiments. The 2SR-GGD method also showsimprovement over GGD detection. What's more, the presented algorithmgenerally needs fewer iterations than GGD_ID discussed in Example 1 C toreach similar detection results. Also, the final results of thepresented algorithm are better than GGD_ID.

B. Two Peak SR Noise-Enhanced Gaussian Assumption-Based Detection withMemory (2SR-GBAD-M)

The experiments show the improved performance of the 2SR-GBAD-Malgorithm. In this section, its efficiency and robustness is furtherimproved by introducing memory in the detection enhancement scheme.

To find the optimum SR noise, the exact knowledge of the probabilitydistribution under the two hypotheses and the determination of thesolution for a set of equations are required. However, in real-worldapplications, due to incomplete information about the distribution,limitations on the accuracy when solving the equations and variouscontents of mammograms, high efficiency and robustness of the SRnoise-enhanced detection system may not be achievable using the SRnoise-based enhancement procedure only once. Multiple applications ofthe procedure may yield further enhancement of detection performance.Therefore, suitably arranged multiple two peak SR noises are appliedmultiple times to increase the efficiency and robustness of thedetection system, which is referred to as 2SR-GBAD-M.

Formally, for the SR noise-based scheme with memory, there isz= y+ n*  (14)where n* represents multiple-peak SR noises instead of a single two peakSR noise added to the original mammogram data in Step 4 of the algorithmpresented in Example 3A, and

$\begin{matrix}{p_{{\overset{\_}{n}}^{*}} = {\sum\limits_{k = 1}^{r}{{w(k)}p_{{\overset{\_}{n}}_{k}}}}} & (15)\end{matrix}$where w_(k) is the weight or probability of occurrence of the k^(th)two-peak SR noise, 0≦w_(k)≦1 and

${\sum\limits_{k = 1}^{r}w_{k}} = 1.$r is the number of the two-peak SR noises which in our current workequals the number of iterations already run plus 1 (i.e., the SR noisedetermined from the estimated probability mass function (PMF) and theupdated threshold at current iteration is also included, where PMF isused as the specific form of the probability distribution for discretedigital mammogram data), andp _(n) _(k) ( n )=λ_(k)δ( n− n _(1k))+(1−λ_(k))δ( n− n _(2k))  (16)Of course, we can change the memory size by using different values of r,but in any case the latest r two-peak SR noises should be employed. Whenr=1, a single two-peak noise is used, and the scheme reduces to thescheme without memory.

At each step of iteration, a larger weight, i.e., higher probability isallocated to the SR noise calculated from the currently estimated PMFs,and the weights for the rest of the SR noises are inversely proportionalto the distances between their corresponding PMFs and the currentlyestimated ones. The distance D between the PMFs obtained during thei^(th) iteration and the latest estimated PMFs is defined as

$\begin{matrix}{D_{l} = {\sum\limits_{i = 0}^{255}\left\lbrack {{{{{PMF}_{{lH}_{o}}(i)} - {{PMF}_{{eH}_{o}}(i)}}} + {{{{PMF}_{{lH}_{1}}(i)} - {{PMF}_{{eH}_{1}}(i)}}}} \right\rbrack}} & (17)\end{matrix}$where PMF_(lH) _(j) denotes the PMF under hypothesis H_(j) obtainedduring the l^(th) iteration, and j=0, 1. PMF_(eH) _(j) is the estimatedPMF under hypothesis H_(j) obtained at the current iteration. Thesummation is over all possible image intensity values. This approach toincorporate memory has resulted in encouraging results as will be seenin the Example 5, infra. The detection results of the two peak SR noiseenhanced GBAD tests with memory are shown in FIGS. 13 (f), 14 (f) and 15(f), from which the Gaussian assumption-based detection can be seensuffering from model mismatch is improved through the addition of SRnoise. Experiments also show that 2SR-GGD-M yields better performancethan 2SR-GBAD-M. A more general SR noise-based detection enhancementframework based on the work in this Example is presented in Example 4,infra.

Example 4 SR Noise-Based Detection Enhancement Framework

A SR noise-based detection enhancement method was presented in Example 3to reduce the model mismatch resulting from the Gaussian assumption.When models other than Gaussian models are used to fit data, there maystill exist model mismatches, resulting in detector performancedegradation, and SR noise may enhance the detector performance. In thisExample, the SR noise-based detection scheme is extended and a moregeneral SR noise-based detection enhancement framework is presented.This framework provides much more flexibility and higher efficiency. Thedetectors (or classifiers) which are controlled are the ones that areconsidered herein, i.e., their thresholds can be changed.

The framework is developed by modifying the first two steps of thedetection procedure presented in Example 3 and is shown as follows.

Initialization: Initial detection.

-   Step 1: Probability density estimates are obtained under the two    hypotheses using the detected positive (lesion) and negative    (background) pixels. The detection threshold (or the classifier) is    updated according to the estimated probability density information.-   Step 2: The pixels are classified (or detected) with the updated    threshold or the classifier in Step 1. The resulting detected    positive and negative pixels are employed for estimating probability    densities under the two hypotheses.-   Step 3: The updated threshold or classifier in Step 1 and the newly    estimated probability densities in Step 2 are used to determine the    SR noise with the method mentioned in Example 2.-   Step 4: The mammogram data is modified by adding SR noise to the    original pixel intensities.-   Step 5: Detection is performed with the detector or classifier    updated in Step 1 using the modified data from Step 4. If the    difference between two successive detection results is very small,    terminate the algorithm else go to Step 1.

To improve the efficiency and robustness of the detection framework, thetwo peak SR noise scheme with memory, which yields multi peak SR noise,can also be used in Step 4.

It is noted that no specific constraints are put on the initialization,threshold or classifier updating and PDF estimation methods used in thisframework. Any reasonable approaches could be employed. As shown herein,the ability of the framework is illustrated by considering differentalgorithms for threshold or classifier updating and PDF estimation. Forinitialization, the maxima filter discussed in Example 1A is still used.For threshold or classifier updating, one may use the methods that canconverge when there is no SR noise added, such as GMDC and iterativemode separation (IMS) algorithms. IMS is an unsupervised learningpattern classification approach, which employs kernel density estimationtechnique to determine the PDF and performs clustering in an iterativemanner. For PDF estimation, one may use non-parametric methods, such askernel density estimation, k-nearest neighbor density estimation andBootstrap methods, etc., because it is desirable to reduce the modelmismatch during the PDF estimation as well as the dependence of theframework on modeling, and to make the framework more generally usable.Efron, B. and R. J. Tibshirani, An Introduction to the Bootstrap,Chapman & Hall, 1993; Jose I. De la Rosa and Gilles A. Fleury,“Bootstrap Methods for a Measurement Estimation Problem,” IEEE Trans. onInstrumentation and Measurement, vol. 55, no. 3, pp. 820-827, June,2006. In accordance with an embodiment of the present invention, forperformance comparison, the kernel density estimation approach andthreshold update using Eq. (18) is employed, same as those used in IMS.It is observed in Example 5 that the SR noise-based method can furtherimprove the performance of IMS. The threshold updating is carried out byusingP ₀ p ₀(y*)=P ₁ p ₁(y*)  (18)where y* is the updated detection threshold during the currentiteration. P₀ and P₁ are the a priori probabilities of the detectednegatives and positives, which can be estimated by {circumflex over(P)}_(i)=n_(i)/n, where n_(i) is the number of negatively detectedpixels when i=0 and positively detected pixels when i=1, and n is thetotal number of pixels in the mammogram. This generates a suboptimaldetector because the threshold is determined from the coarsely estimateda priori probabilities and PDFs by using the plug-in rule. GeoffreyMclachlan and David Peel, Finite Mixture Models, A Wiley-IntersciencePublication, 2, October, 2000.

Experimental results show that the SR noise-based algorithm presentedherein generally needs fewer number of iterations than IMS to reachsimilar detection results. Also, the final results of SR noise-basedalgorithm are better than IMS, where the final results are attained whenthe difference between two successive detection results is very small.In addition, given a good initialization, satisfactory detection canstill be obtained even when the threshold or classifier update procedurein Step 1 is not performed during the iterations.

It can be seen that the above iterative procedure includes a scheme forPDF estimation, but in the current detection (or clustering)application, the estimated PDFs are not of interest as an end inthemselves. Instead, the detection results are the focus here which, ofcourse, depend on the estimate. At the same time, an accurate PDFestimate can also be obtained from an accurate detection. So, thedetection results are used in this paper as an alternative way toevaluate the performance of the PDF estimation algorithm. Forcomparison, a Gaussian mixture modeling (GMM)-based clustering methodwhich performs the detection based on the GMM fitted pdf is employedwith the detection results shown in Example 5, infra.

Example 5 Performance Comparison and Analysis

This Example relates to the presentation of experimental results and theperformance evaluation of several lesion detection and classificationalgorithms. The algorithms that are compared and analyzed include GBAD,GGD and GGD_ID discussed in Example 1, GADC, 2SR-GBAD-M, IMS, GMM-basedclustering method, high order statistics method based on local maximadetection and adaptive wavelet transform (HOSLW) and the SR noise-baseddetection enhancement framework using a procedure similar to IMS, i.e.,SR_IMS.

The first four algorithms are based on the Gaussian distributionassumption and are parametric approaches. GMM is a semi-parametrictechnique for PDF estimation, in which the superposition of a number ofparametric densities, e.g., Gaussian distribution, are used toapproximate the underlying PDF. It offers a useful compromise betweenthe non-parametric methods mentioned in Example 4, and the parametricestimation methods, such as those mentioned above. For the clusteringapplication, the GMM given in Eq. (19) is first fit by using theExpectation-maximization algorithm

$\begin{matrix}{{f(y)} = {\sum\limits_{i = 1}^{g}{P_{i}{f_{i}(y)}}}} & (19)\end{matrix}$where f(y) is the density of the observation y, and f_(i)(y) are thecomponent densities of the mixture. g is the number of components, whichcan be preset or automatically determined according to the datastatistics. As discussed herein, we set g=2 to facilitate two-classclustering. P_(i) are the mixing proportions or weights, 0≦P_(i)≦1 (i=1,. . . , g) and

${\sum\limits_{i = 1}^{g}P_{i}} = 1.$

The clustering is performed by using the plug-in rule given in Eq. (20)based on the Bayes ruleR(y)=i if {circumflex over (P)} _(i) {circumflex over (f)}_(i)(y)≧{circumflex over (P)} _(j) {circumflex over (f)} _(j)(y)  (20)for j=1, . . . , g, where R(y)=i denotes that the allocation rule R(y)assigns the observation y to the i^(th) component of the mixture model.{circumflex over (P)}_(i) and {circumflex over (f)}_(i)(y) are thefitted values of P_(i) and f_(i)(y), respectively.

The HOSLW algorithm is proved to have superior performance compared withother existing methods in terms of efficiency and reliability. In thismethod, local maxima of the mammogram are determined as the lesioncandidates, and the adaptive wavelet transform is employed to generatesubbands which permit the rank of these maxima in the subband mammogramusing a higher order statistical test for lesion detection.

For fairness, the same initial detection is used for the algorithmscompared in the experiments. In 2SR-GBAD-M and SR_IMS, the weights ofthe two peak SR noise calculated from the currently estimated pdfs areset to be 0.5.

Experiments have been carried out using 75 images, and the results offive detection/classification algorithms are presented in Table 2,infra, in terms of true-positive fraction (TP) and false positives perimage (FPI), where TP is defined as the ratio of the number of the truepositive marks to the number of lesions and FPI is defined as theaverage number of false positives per image. In these experiments, if adetected positive area has more than 50% overlap with the ground trutharea, the detected area is considered to be a TP lesion. Otherwise, itis considered to be a false positive. This is the same definition asused in Heng-Da Cheng, Yui Man Lui, and Rita I. Freimanis, “A novelapproach to micro-calcification detection using fuzzy logic technique,”IEEE Trans. on Medical Imaging, vol. 17, no. 3, June 1998.

The qualitative evaluation of these algorithms are presented first.FIGS. 13, 14 and 15 show the experimental results for the three ROIs cutfrom three representative mammograms, where the detected positive pixelsare labeled with small dots. In brief, FIG. 13 shows original abnormalmammogram and the detection results (Abnormal mammogram type 1:homogeneous background with small number of isolatedmicro-calcifications). (a) Original mammogram with micro-calcifications;(b) GBAD; (c) GGD; (d) GGD_ID; (e) GADC; (f) 2SR-GBAD-M; (g) IMS; (h)HOSLW; (i) GMM-based detection; (j) SR_IMS. FIG. 14 shows originalabnormal mammogram and the detection results (Abnormal mammogram type 2:homogeneous background with large number of micro-calcifications(clusters)). (a) Original mammogram with micro-calcifications; (b) GBAD;(c) GGD; (d) GGD_ID; (e) GADC; (f) 2SR-GBAD-M; (g) IMS; (h) HOSLW; (i)GMM-based detection; (j) SR_IMS. FIG. 15 shows original abnormalmammogram and the detection results (Abnormal mammogram type 3:inhomogeneous background with moderate number of micro-calcifications(clusters)). (a) Original mammogram with micro-calcifications; (b) GBAD;(c) GGD; (d) GGD_ID; (e) GADC; (f) 2SR-GAD-M; (g) IMS; (h) HOSLW; (i)GMM-based detection; (j) SR_IMS. For each Figure, the detected positivepixels are labeled with dots.

In the experiment shown in FIG. 13, a fixed threshold is employed in the2SR-GBAD-M and SR_IMS algorithms throughout the iterations. Thecomplexity of the mammogram used in these experiments is the lowestcompared with the other two to be discussed next. From the figures, itcan be seen that the GBAD and GGD methods find all the lesions, but atthe same time generate many false alarms (see 13(b) and 13(c)). GGD_ID(13(d)) is a more robust method. It improves the detection of GGD andperforms better than the GADC and IMS methods shown in 13(e) and 13(g),but it still fails to reduce the false positives satisfactorily. Theadvantage of the GADC is that it converges quickly, generally in no morethan 8 iterations in these experiments, while IMS converge to localextrema. HOSLW method (13(h)) can find the lesions efficiently, but itfails to determine lesion shape which plays a very important role indiscriminating the benign tumors from the malignant ones. Moreover, itsdetection performance depends on how accurately the number of lesionpixels can be estimated, which is generally not available or known inreal-world cases. These detectors suffer from model mismatch andparameter suboptimality resulting in suboptimum detection threshold, andtheir performances are degraded. The GMM-based detector finds all thelesions but still does not avoid the high P_(F) (see 2(i)), which is dueto the inaccuracy when GMM is used to fit the mammogram data. Incontrast, the presented 2SR-GBAD-M and SR_IMS algorithms yield gooddetection results in terms of lesion localization, lesion contourexploration and P_(F) reduction (see 13(f) and 13(j)), whichdemonstrates the capability of the SR noise-based method for enhancingthe detectors with model mismatch and parameter suboptimality. Comparing13(f) and 13(j), it can be seen that SR_IMS performs a little betterthan 2SR-GBAD-M in reducing false alarms and determining lesionboundaries.

FIG. 14 shows a more complex case, where both isolatedmicro-calcifications and crowded clusters exist and the number oflesions is large. It can be seen that still the 2SR-GBAD-M and SR_IMSalgorithms yield better detections with clearer lesion contours and lessfalse positives (see 14(f) and 14(j)). Compared with GBAD and GGD in14(b) and 14(c), GGD_ID and GADC method shown in 14(d) and 14(e) performbetter but still with high P_(F)'s. IMS fails to find some lesions (see14(g)). HOSLW (14(h)) does not provide much useful information about thelesion positions in this crowded micro-calcifications (clusters) case.This is because its detection operation is performed in subband imageswhich have a quarter of the size of the original mammogram, so the areaof the detected positives will be four times of those in the subbandimages when the detection result is shown in the original mammogram.When the micro-calcifications (clusters) are close to each other, theirboundaries and locations are hard to determine. GMM performs better thanthe rest of the methods (except for 14(f) and 14(j)), but stillgenerates many false alarms.

FIG. 15 is the most complex case, where the background distribution isinhomogeneous and some background pixels have their intensitiesapproaching the lesion pixels. It is hard to model the background usingjust a univariate probability distribution. Finite mixture models may bea choice, but to determine the model type and parameters is also achallenging task. Also, their performance could be deteriorated by thenon-stationary nature of the images. Therefore, model mismatch in thistype of images is more serious and unavoidable. In this experiment,univariate Gaussian distribution is still used to model the pixelintensity distributions of the background and lesion, respectively,through which the model mismatch is simulated. From FIG. 15, it can beseen that the performance of all the detectors degrades to some extentwith higher P_(F) and lower P_(D) values as well as more impreciselesion contours compared with the previous two cases. But the presented2SR-GBAD-M and SR_IMS algorithms (see FIG. 15( f) and FIG. 15( j)) stillstand out with better detection results, which again demonstrate theirefficiency in reducing the negative influences of model mismatch andsuboptimum parameters.

Next, the quantitative measurement of the algorithms is shown. Threemethods are selected to compare with 2SR-GBAD-M and SR_IMS, and theresults are presented in Table 2. The reason GADC and IMS were chosen isthat they are all classical pattern classification methods and alsobased on iterative procedures, like 2SR-GBAD-M and SR_IMS. GADC maysuffer from model mismatch due to the Gaussian assumption and IMS mayhave suboptimum threshold value due to the inaccuracy of the pdfestimation when processing mammogram data. Additionally, HOLSW is saidto be superior to several micro-calcification detectors.

TABLE 2 DETECTION PERFORMANCE OF FIVE ALGORITHMS METHODS RESULTS GADCIMS 2SR-G SR_I HO Range [0.61, 1]    [0.58, 1]    [0.80, 1]   [0.81,1]   [0.81, 1]    TP Mean 0.89 0.90 0.93 0.94 0.94 Standard 0.25 0.280.12 0.11 0.11 deviation Range [0, 20] [0, 17] [0, 9] [0, 7] [0, 14] FPIMean 8.16 7.89 4.91 3.12 5.22 Standard 6.18 7.08 3.94 2.95 4.82deviation 2SR-G: 2SR-GBAD-M; SR_I: SR_IMS; HO: HOLSW.

TP and FPI are employed as the metrics. The means of TP and FPIrepresent the average performance of each method, and their standarddeviations are used as a measure of the robustness of each method whenapplied to different types of images. A better method is identified tobe one with higher mean TP value but lower mean FPI value as well aslower TP and FPI standard deviations.

Since HOLSW requires the knowledge of the number of lesions, which isgenerally not available in real-world applications, the lesion number isadjusted manually, such that the TPs of HOLSW and SR_IMS for each imageare the same, and then FPI is employed as a criterion for theirperformance comparison.

From Table 2, it can be seen that 2SR-GBAD-M and SR_IMS achieve superiorperformance than the classical methods, GADC and IMS, both in truepositive detection and in false positive reduction. HOLSW can attain thesimilar true positive detection performance to 2SR-GBAD-M and SR_IMS,but it is worse than the two SR noise-enhanced detectors in terms of FPIreduction. 2SR-GBAD-M and SR_IMS have similar detection results,(actually SR_IMS performs a little better) but SR_IMS yields moresatisfactory results in terms of FPI reduction. This is because2SR-GBAD-M updates the threshold based on the Gaussian assumption, andis therefore affected by the model mismatch.

It should be emphasized that the detection performance of the detectorsmay be further improved if image enhancement techniques are employedbefore detection and post-processing methods, such as patternclassifiers embedded with other lesion features, are used after thedetection procedure.

In summary, automatic detection techniques for micro-calcifications arevery important for breast cancer diagnosis and treatment. Therefore, itis imperative that the detection techniques be developed that detectmicro-calcifications accurately. The experiments described herein showthe development of a lesion detection approach based on SR noise forenhancing the Gaussian assumption-based detectors which suffer frommodel mismatch, and furthermore show a more general SR noise-baseddetection enhancement framework. Comparative performance evaluation wascarried out via experiments between the presented SR noise-baseddetection enhancement schemes and several detection and classificationtechniques with three types of representative abnormal mammograms. Theresults show that the presented algorithm and the framework resulted inhighly encouraging performance in terms of flexibility, detectionefficiency and system robustness, which demonstrates the SR noise'scapability of enhancing the suboptimal detectors and supports itsreal-world CAD application.

There are many other proposed embodiments of the present invention.Optimizing the SR noise-based technique with memory described hereinwill be very useful to further improve the efficiency and robustness ofthe SR noise-based detection enhancement scheme, by determining theoptimum weights for two-peak SR noises. Extension of the SR noise-basedtechnique to enhancing fixed multiple threshold detectors is alsocontemplated. The performance of SR enhanced variable detectors has beenshown to be superior than the fixed ones, where both the SR noise andthe critical function can be jointly designed to enhance detection. SoSR noise-based detectors incorporating variable critical function arelikely to be promising. In the experiments discussed herein, the casewhere signal and background noise are all independently distributed isconsidered. Additional embodiments covering the correlated signal andnoise case may further improve the detection performance. In addition,the application of the detection schemes developed and described hereinto other two types of mammogram lesions, i.e., mass and spiculatedlesions, and even other medical images, is contemplated. Finally, the SRnoise-enhanced scheme may also be useful in color images, which could bean excellent extension of embodiments of the present invention to morereal world applications.

While several embodiments of the invention have been discussed, it willbe appreciated by those skilled in the art that various modificationsand variations of the present invention are possible. Such modificationsdo not depart from the spirit and scope of the claimed invention.

1. A method of increasing the probability of detecting at least onemicro-calcification lesion in a mammogram, said method comprising thesteps of: a. obtaining digital mammogram data by using an X-ray digitalmammography system; b. calculating stochastic resonance noise with aprocessor by determining the stochastic resonance noise probabilitydensity function that does not increase the probability of false alarmand calculating the stochastic resonance noise data probability densityfunction from a known probability density function for said digitalmammogram data, wherein the stochastic resonance noise probabilitydensity function equals p_(n) ^(opt)(n)=λδ(n−n₁)+(1−λ)δ(n−n₂), withvalues n₁ and n₂ equal to two delta function locations havingprobabilities of λ and 1−λ respectively, where n₁ and n₂ denote twodiscrete vectors, λ denotes the probability of occurrence of said twovectors, and n is a randomization of said two discrete vectors addedwith the probabilities of λ and 1−λ, respectively; and c. adding saidstochastic resonance noise to said digital mammogram data by using saidprocessor, thereby improving the detection of said at least onemicro-calcification lesion in the mammogram.
 2. The method of claim 1,wherein the step of obtaining further comprises the step of performingcoarse detection of said at least one micro-calcification lesion in themammogram using a local maxima filter.
 3. The method of claim 2, furthercomprising the step of calculating the detection threshold for aGaussian background assumption-based detector.
 4. The method of claim 3,further comprising the step of estimating mean μ_(b) and variance σ_(b)² of the background based on detected negative pixels.
 5. The method ofclaim 4, updating the detection threshold for the Gaussian backgroundassumption-based detector.
 6. The method of claim 5, wherein the step ofupdating the detection threshold is based on the desired P_(F), meanμ_(b) and variance σ_(b) ² using the following equation:${P_{F} = {{\int_{\{{\overset{\_}{y}:{{L{(\overset{\_}{y})}} > \gamma}}}^{\;}{{p\left( {\overset{\_}{y}\text{;}H_{0}} \right)}\ {\mathbb{d}\overset{\_}{y}}}} = \beta}},$where β is the desired value of the P_(F), and p( y; H₀) is a Gaussianprobability density function under hypothesis H₀: y[m]=w[m], where m isa pixel index corresponding to a pixel observation under consideration,y[m] is intensity of said pixel observation under consideration, largerthan or equal to zero, and w[m] is background noise that is assumed toobey Gaussian distribution with mean μ_(b) and variance σ_(b) ².
 7. Themethod of claim 5, further comprising the step of detecting positive andnegative pixels with said updated detection threshold.
 8. The method ofclaim 7, further comprising the step of estimating probability densitiesusing kernel density estimation method based on the resulting detectedpositive and negative pixels under the following two hypotheses:H ₀ : y[m]=w[m]H ₁ : y[m]=s[m]+w[m] where m is a pixel index corresponding to a pixelobservation under consideration, y[m] is intensity of said pixelobservation under consideration, larger than or equal to zero, s[m] islesion signal, and w[m] is background noise that is assumed to obeyGaussian distribution with mean μ_(b) and variance σ_(b) ².
 9. Themethod of claim 8, wherein the step of calculating stochastic resonancenoise further comprises the step of using the updated detectionthreshold and the estimated probability densities to calculate thestochastic resonance noise.
 10. The method of claim 9, furthercomprising the step of performing the step of detection on thestochastic resonance noise modified digital mammogram data.
 11. Themethod of claim 10, further comprising the step of determining thedifference between the course detection results and the results obtainedfrom the detection performed on the stochastic resonance noise modifieddigital mammogram data.
 12. The method of claim 11, repeating thefollowing steps in an iterative manner: estimating mean μ_(b) andvariance σ_(b) ² the background based on detected negative pixels;updating the detection threshold for the Gaussian backgroundassumption-based detector; detecting positive and negative pixels withsaid updated detection threshold; estimating probability densities;calculating stochastic resonance noise; modifying said digital mammogramdata by adding said stochastic resonance noise to said digital mammogramdata; and performing the step of detection on the stochastic resonancenoise modified digital mammogram data.
 13. The method of claim 1,further comprising the step of introducing memory into said method byapplying multiple two-peak stochastic resonance noises multiple times.14. The method of claim 13, wherein the step of introducing memory intosaid method by applying multiple two-peak stochastic resonance noisesmultiple times further comprises the step of using the followingmultiple two-peak stochastic resonance noise scheme:z= y+ n*, where n* represents multiple-peak stochastic resonance noisesadded to the original mammogram data, where z denotes the originalmammogram data modified through adding the stochastic resonance noise tothe original mammogram data and y denotes the original mammogram data,and$p_{{\overset{\_}{n}}^{*}} = {\sum\limits_{k = 1}^{r}\;{w_{k}p_{{\overset{\_}{n}}_{k}}}}$where P _(n) * denotes the probability mass function of n*, k is theindex of the two-peak SR noise, w_(k) is the weight or probability ofoccurrence of the k^(th) multiple-peak stochastic resonance noise,${{0 \leq w_{k} \leq {1\mspace{14mu}{and}\mspace{14mu}{\sum\limits_{k = 1}^{r}\; w_{k}}}} = 1},$and r is the number of the two-peak stochastic resonance noises, andp _(n) _(k) ( n )=λ_(k)δ( n− n _(1k))+(1−λ_(k))δ( n− n _(2k)).